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1095 lines
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">A. Find The Array</div><div class="time-limit"><div class="property-title">time limit per test</div>1 second</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file input-standard"><div class="property-title">input</div>standard input</div><div class="output-file output-standard"><div class="property-title">output</div>standard output</div></div><div><p>Let's call an array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-2">a</span></span></span><script type="math/tex" id="MathJax-Element-1">a</script> consisting of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-3"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-4">n</span></span></span><script type="math/tex" id="MathJax-Element-2">n</script> positive (greater than <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-5"><span class="MJXp-mn" id="MJXp-Span-6">0</span></span></span><script type="math/tex" id="MathJax-Element-3">0</script>) integers <span class="tex-font-style-it">beautiful</span> if the following condition is held for every <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-7"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-8">i</span></span></span><script type="math/tex" id="MathJax-Element-4">i</script> from <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-9"><span class="MJXp-mn" id="MJXp-Span-10">1</span></span></span><script type="math/tex" id="MathJax-Element-5">1</script> to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-11"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-12">n</span></span></span><script type="math/tex" id="MathJax-Element-6">n</script>: either <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-13"><span class="MJXp-msubsup" id="MJXp-Span-14"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-15" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-16" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-17" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-18">1</span></span></span><script type="math/tex" id="MathJax-Element-7">a_i = 1</script>, or at least one of the numbers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-19"><span class="MJXp-msubsup" id="MJXp-Span-20"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-21" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-22" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-23" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-24">1</span></span></span><script type="math/tex" id="MathJax-Element-8">a_i - 1</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-25"><span class="MJXp-msubsup" id="MJXp-Span-26"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-27" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-28" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-29" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-30">2</span></span></span><script type="math/tex" id="MathJax-Element-9">a_i - 2</script> exists in the array as well.</p><p>For example: </p><ul><li> the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-31"><span class="MJXp-mo" id="MJXp-Span-32" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-33">5</span><span class="MJXp-mo" id="MJXp-Span-34" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-35">3</span><span class="MJXp-mo" id="MJXp-Span-36" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-37">1</span><span class="MJXp-mo" id="MJXp-Span-38" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-10">[5, 3, 1]</script> is beautiful: for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-39"><span class="MJXp-msubsup" id="MJXp-Span-40"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-41" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-42" style="vertical-align: -0.4em;">1</span></span></span></span><script type="math/tex" id="MathJax-Element-11">a_1</script>, the number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-43"><span class="MJXp-msubsup" id="MJXp-Span-44"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-45" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-46" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-47" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-48">2</span><span class="MJXp-mo" id="MJXp-Span-49" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-50">3</span></span></span><script type="math/tex" id="MathJax-Element-12">a_1 - 2 = 3</script> exists in the array; for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-51"><span class="MJXp-msubsup" id="MJXp-Span-52"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-53" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-54" style="vertical-align: -0.4em;">2</span></span></span></span><script type="math/tex" id="MathJax-Element-13">a_2</script>, the number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-55"><span class="MJXp-msubsup" id="MJXp-Span-56"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-57" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-58" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-59" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-60">2</span><span class="MJXp-mo" id="MJXp-Span-61" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-62">1</span></span></span><script type="math/tex" id="MathJax-Element-14">a_2 - 2 = 1</script> exists in the array; for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-63"><span class="MJXp-msubsup" id="MJXp-Span-64"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-65" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-66" style="vertical-align: -0.4em;">3</span></span></span></span><script type="math/tex" id="MathJax-Element-15">a_3</script>, the condition <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-67"><span class="MJXp-msubsup" id="MJXp-Span-68"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-69" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-70" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-71" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-72">1</span></span></span><script type="math/tex" id="MathJax-Element-16">a_3 = 1</script> holds; </li><li> the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-73"><span class="MJXp-mo" id="MJXp-Span-74" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-75">1</span><span class="MJXp-mo" id="MJXp-Span-76" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-77">2</span><span class="MJXp-mo" id="MJXp-Span-78" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-79">2</span><span class="MJXp-mo" id="MJXp-Span-80" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-81">2</span><span class="MJXp-mo" id="MJXp-Span-82" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-83">2</span><span class="MJXp-mo" id="MJXp-Span-84" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-17">[1, 2, 2, 2, 2]</script> is beautiful: for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-85"><span class="MJXp-msubsup" id="MJXp-Span-86"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-87" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-88" style="vertical-align: -0.4em;">1</span></span></span></span><script type="math/tex" id="MathJax-Element-18">a_1</script>, the condition <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-89"><span class="MJXp-msubsup" id="MJXp-Span-90"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-91" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-92" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-93" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-94">1</span></span></span><script type="math/tex" id="MathJax-Element-19">a_1 = 1</script> holds; for every other number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-95"><span class="MJXp-msubsup" id="MJXp-Span-96"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-97" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-98" style="vertical-align: -0.4em;">i</span></span></span></span><script type="math/tex" id="MathJax-Element-20">a_i</script>, the number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-99"><span class="MJXp-msubsup" id="MJXp-Span-100"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-101" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-102" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-103" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-104">1</span><span class="MJXp-mo" id="MJXp-Span-105" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-106">1</span></span></span><script type="math/tex" id="MathJax-Element-21">a_i - 1 = 1</script> exists in the array; </li><li> the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-107"><span class="MJXp-mo" id="MJXp-Span-108" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-109">1</span><span class="MJXp-mo" id="MJXp-Span-110" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-111">4</span><span class="MJXp-mo" id="MJXp-Span-112" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-22">[1, 4]</script> is not beautiful: for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-113"><span class="MJXp-msubsup" id="MJXp-Span-114"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-115" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-116" style="vertical-align: -0.4em;">2</span></span></span></span><script type="math/tex" id="MathJax-Element-23">a_2</script>, neither <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-117"><span class="MJXp-msubsup" id="MJXp-Span-118"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-119" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-120" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-121" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-122">2</span><span class="MJXp-mo" id="MJXp-Span-123" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-124">2</span></span></span><script type="math/tex" id="MathJax-Element-24">a_2 - 2 = 2</script> nor <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-125"><span class="MJXp-msubsup" id="MJXp-Span-126"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-127" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-128" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-129" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-130">1</span><span class="MJXp-mo" id="MJXp-Span-131" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-132">3</span></span></span><script type="math/tex" id="MathJax-Element-25">a_2 - 1 = 3</script> exists in the array, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-133"><span class="MJXp-msubsup" id="MJXp-Span-134"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-135" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-136" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-137" style="margin-left: 0.333em; margin-right: 0.333em;">≠</span><span class="MJXp-mn" id="MJXp-Span-138">1</span></span></span><script type="math/tex" id="MathJax-Element-26">a_2 \ne 1</script>; </li><li> the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-139"><span class="MJXp-mo" id="MJXp-Span-140" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-141">2</span><span class="MJXp-mo" id="MJXp-Span-142" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-27">[2]</script> is not beautiful: for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-143"><span class="MJXp-msubsup" id="MJXp-Span-144"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-145" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-146" style="vertical-align: -0.4em;">1</span></span></span></span><script type="math/tex" id="MathJax-Element-28">a_1</script>, neither <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-147"><span class="MJXp-msubsup" id="MJXp-Span-148"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-149" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-150" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-151" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-152">1</span><span class="MJXp-mo" id="MJXp-Span-153" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-154">1</span></span></span><script type="math/tex" id="MathJax-Element-29">a_1 - 1 = 1</script> nor <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-155"><span class="MJXp-msubsup" id="MJXp-Span-156"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-157" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-158" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-159" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-160">2</span><span class="MJXp-mo" id="MJXp-Span-161" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-162">0</span></span></span><script type="math/tex" id="MathJax-Element-30">a_1 - 2 = 0</script> exists in the array, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-163"><span class="MJXp-msubsup" id="MJXp-Span-164"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-165" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-166" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-167" style="margin-left: 0.333em; margin-right: 0.333em;">≠</span><span class="MJXp-mn" id="MJXp-Span-168">1</span></span></span><script type="math/tex" id="MathJax-Element-31">a_1 \ne 1</script>; </li><li> the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-169"><span class="MJXp-mo" id="MJXp-Span-170" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-171">2</span><span class="MJXp-mo" id="MJXp-Span-172" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-173">1</span><span class="MJXp-mo" id="MJXp-Span-174" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-175">3</span><span class="MJXp-mo" id="MJXp-Span-176" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-32">[2, 1, 3]</script> is beautiful: for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-177"><span class="MJXp-msubsup" id="MJXp-Span-178"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-179" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-180" style="vertical-align: -0.4em;">1</span></span></span></span><script type="math/tex" id="MathJax-Element-33">a_1</script>, the number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-181"><span class="MJXp-msubsup" id="MJXp-Span-182"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-183" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-184" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-185" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-186">1</span><span class="MJXp-mo" id="MJXp-Span-187" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-188">1</span></span></span><script type="math/tex" id="MathJax-Element-34">a_1 - 1 = 1</script> exists in the array; for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-189"><span class="MJXp-msubsup" id="MJXp-Span-190"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-191" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-192" style="vertical-align: -0.4em;">2</span></span></span></span><script type="math/tex" id="MathJax-Element-35">a_2</script>, the condition <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-193"><span class="MJXp-msubsup" id="MJXp-Span-194"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-195" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-196" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-197" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-198">1</span></span></span><script type="math/tex" id="MathJax-Element-36">a_2 = 1</script> holds; for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-199"><span class="MJXp-msubsup" id="MJXp-Span-200"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-201" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-202" style="vertical-align: -0.4em;">3</span></span></span></span><script type="math/tex" id="MathJax-Element-37">a_3</script>, the number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-203"><span class="MJXp-msubsup" id="MJXp-Span-204"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-205" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-206" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-207" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-208">2</span><span class="MJXp-mo" id="MJXp-Span-209" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-210">1</span></span></span><script type="math/tex" id="MathJax-Element-38">a_3 - 2 = 1</script> exists in the array. </li></ul><p>You are given a positive integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-211"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-212">s</span></span></span><script type="math/tex" id="MathJax-Element-39">s</script>. Find the minimum possible size of a beautiful array with the sum of elements equal to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-213"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-214">s</span></span></span><script type="math/tex" id="MathJax-Element-40">s</script>.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains one integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-215"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-216">t</span></span></span><script type="math/tex" id="MathJax-Element-41">t</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-217"><span class="MJXp-mn" id="MJXp-Span-218">1</span><span class="MJXp-mo" id="MJXp-Span-219" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-220">t</span><span class="MJXp-mo" id="MJXp-Span-221" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-222">5000</span></span></span><script type="math/tex" id="MathJax-Element-42">1 \le t \le 5000</script>) — the number of test cases.</p><p>Then <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-223"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-224">t</span></span></span><script type="math/tex" id="MathJax-Element-43">t</script> lines follow, the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-225"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-226">i</span></span></span><script type="math/tex" id="MathJax-Element-44">i</script>-th line contains one integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-227"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-228">s</span></span></span><script type="math/tex" id="MathJax-Element-45">s</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-229"><span class="MJXp-mn" id="MJXp-Span-230">1</span><span class="MJXp-mo" id="MJXp-Span-231" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-232">s</span><span class="MJXp-mo" id="MJXp-Span-233" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-234">5000</span></span></span><script type="math/tex" id="MathJax-Element-46">1 \le s \le 5000</script>) for the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-235"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-236">i</span></span></span><script type="math/tex" id="MathJax-Element-47">i</script>-th test case.</p></div><div class="output-specification"><div class="section-title">Output</div><p>Print <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-237"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-238">t</span></span></span><script type="math/tex" id="MathJax-Element-48">t</script> integers, the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-239"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-240">i</span></span></span><script type="math/tex" id="MathJax-Element-49">i</script>-th integer should be the answer for the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-241"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-242">i</span></span></span><script type="math/tex" id="MathJax-Element-50">i</script>-th testcase: the minimum possible size of a beautiful array with the sum of elements equal to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-243"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-244">s</span></span></span><script type="math/tex" id="MathJax-Element-51">s</script>.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id0011335133290051413" id="id007175500850450626" class="input-output-copier">Copy</div></div><pre id="id0011335133290051413">4
|
||
1
|
||
8
|
||
7
|
||
42
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id005021691657849819" id="id004270717779247326" class="input-output-copier">Copy</div></div><pre id="id005021691657849819">1
|
||
3
|
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3
|
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7
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</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>Consider the example test:</p><ol><li> in the first test case, the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-245"><span class="MJXp-mo" id="MJXp-Span-246" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-247">1</span><span class="MJXp-mo" id="MJXp-Span-248" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-52">[1]</script> meets all conditions; </li><li> in the second test case, the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-249"><span class="MJXp-mo" id="MJXp-Span-250" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-251">3</span><span class="MJXp-mo" id="MJXp-Span-252" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-253">4</span><span class="MJXp-mo" id="MJXp-Span-254" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-255">1</span><span class="MJXp-mo" id="MJXp-Span-256" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-53">[3, 4, 1]</script> meets all conditions; </li><li> in the third test case, the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-257"><span class="MJXp-mo" id="MJXp-Span-258" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-259">1</span><span class="MJXp-mo" id="MJXp-Span-260" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-261">2</span><span class="MJXp-mo" id="MJXp-Span-262" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-263">4</span><span class="MJXp-mo" id="MJXp-Span-264" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-54">[1, 2, 4]</script> meets all conditions; </li><li> in the fourth test case, the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-265"><span class="MJXp-mo" id="MJXp-Span-266" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-267">1</span><span class="MJXp-mo" id="MJXp-Span-268" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-269">4</span><span class="MJXp-mo" id="MJXp-Span-270" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-271">6</span><span class="MJXp-mo" id="MJXp-Span-272" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-273">8</span><span class="MJXp-mo" id="MJXp-Span-274" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-275">10</span><span class="MJXp-mo" id="MJXp-Span-276" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-277">2</span><span class="MJXp-mo" id="MJXp-Span-278" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-279">11</span><span class="MJXp-mo" id="MJXp-Span-280" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-55">[1, 4, 6, 8, 10, 2, 11]</script> meets all conditions. </li></ol></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">B. Maximum Cost Deletion</div><div class="time-limit"><div class="property-title">time limit per test</div>2 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file input-standard"><div class="property-title">input</div>standard input</div><div class="output-file output-standard"><div class="property-title">output</div>standard output</div></div><div><p>You are given a string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-281"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-282">s</span></span></span><script type="math/tex" id="MathJax-Element-56">s</script> of length <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-283"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-284">n</span></span></span><script type="math/tex" id="MathJax-Element-57">n</script> consisting only of the characters <span class="tex-font-style-tt">0</span> and <span class="tex-font-style-tt">1</span>.</p><p>You perform the following operation until the string becomes empty: choose some <span class="tex-font-style-bf">consecutive</span> substring of <span class="tex-font-style-bf">equal</span> characters, erase it from the string and glue the remaining two parts together (any of them can be empty) in the same order. For example, if you erase the substring <span class="tex-font-style-tt">111</span> from the string <span class="tex-font-style-tt">1<span class="tex-font-style-bf">111</span>10</span>, you will get the string <span class="tex-font-style-tt">110</span>. When you delete a substring of length <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-285"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-286">l</span></span></span><script type="math/tex" id="MathJax-Element-58">l</script>, you get <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-287"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-288">a</span><span class="MJXp-mo" id="MJXp-Span-289" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-290">l</span><span class="MJXp-mo" id="MJXp-Span-291" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-292">b</span></span></span><script type="math/tex" id="MathJax-Element-59">a \cdot l + b</script> points.</p><p>Your task is to calculate the maximum number of points that you can score in total, if you have to make the given string empty.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-293"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-294">t</span></span></span><script type="math/tex" id="MathJax-Element-60">t</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-295"><span class="MJXp-mn" id="MJXp-Span-296">1</span><span class="MJXp-mo" id="MJXp-Span-297" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-298">t</span><span class="MJXp-mo" id="MJXp-Span-299" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-300">2000</span></span></span><script type="math/tex" id="MathJax-Element-61">1 \le t \le 2000</script>) — the number of testcases.</p><p>The first line of each testcase contains three integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-301"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-302">n</span></span></span><script type="math/tex" id="MathJax-Element-62">n</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-303"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-304">a</span></span></span><script type="math/tex" id="MathJax-Element-63">a</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-305"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-306">b</span></span></span><script type="math/tex" id="MathJax-Element-64">b</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-307"><span class="MJXp-mn" id="MJXp-Span-308">1</span><span class="MJXp-mo" id="MJXp-Span-309" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-310">n</span><span class="MJXp-mo" id="MJXp-Span-311" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-312">100</span><span class="MJXp-mo" id="MJXp-Span-313" style="margin-left: 0em; margin-right: 0.222em;">;</span><span class="MJXp-mo" id="MJXp-Span-314" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-315">100</span><span class="MJXp-mo" id="MJXp-Span-316" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-317">a</span><span class="MJXp-mo" id="MJXp-Span-318" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-319">b</span><span class="MJXp-mo" id="MJXp-Span-320" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-321">100</span></span></span><script type="math/tex" id="MathJax-Element-65">1 \le n \le 100; -100 \le a, b \le 100</script>) — the length of the string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-322"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-323">s</span></span></span><script type="math/tex" id="MathJax-Element-66">s</script> and the parameters <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-324"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-325">a</span></span></span><script type="math/tex" id="MathJax-Element-67">a</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-326"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-327">b</span></span></span><script type="math/tex" id="MathJax-Element-68">b</script>.</p><p>The second line contains the string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-328"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-329">s</span></span></span><script type="math/tex" id="MathJax-Element-69">s</script>. The string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-330"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-331">s</span></span></span><script type="math/tex" id="MathJax-Element-70">s</script> consists only of the characters <span class="tex-font-style-tt">0</span> and <span class="tex-font-style-tt">1</span>.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each testcase, print a single integer — the maximum number of points that you can score.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id007675683682124629" id="id0006505631406218215" class="input-output-copier">Copy</div></div><pre id="id007675683682124629">3
|
||
3 2 0
|
||
000
|
||
5 -2 5
|
||
11001
|
||
6 1 -4
|
||
100111
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id001322358127831661" id="id00996265197942917" class="input-output-copier">Copy</div></div><pre id="id001322358127831661">6
|
||
15
|
||
-2
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first example, it is enough to delete the entire string, then we will get <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-332"><span class="MJXp-mn" id="MJXp-Span-333">2</span><span class="MJXp-mo" id="MJXp-Span-334" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mn" id="MJXp-Span-335">3</span><span class="MJXp-mo" id="MJXp-Span-336" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-337">0</span><span class="MJXp-mo" id="MJXp-Span-338" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-339">6</span></span></span><script type="math/tex" id="MathJax-Element-71">2 \cdot 3 + 0 = 6</script> points.</p><p>In the second example, if we delete characters one by one, then for each deleted character we will get <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-340"><span class="MJXp-mo" id="MJXp-Span-341" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-342" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-343">2</span><span class="MJXp-mo" id="MJXp-Span-344" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-345" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mn" id="MJXp-Span-346">1</span><span class="MJXp-mo" id="MJXp-Span-347" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-348">5</span><span class="MJXp-mo" id="MJXp-Span-349" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-350">3</span></span></span><script type="math/tex" id="MathJax-Element-72">(-2) \cdot 1 + 5 = 3</script> points, i. e. <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-351"><span class="MJXp-mn" id="MJXp-Span-352">15</span></span></span><script type="math/tex" id="MathJax-Element-73">15</script> points in total.</p><p>In the third example, we can delete the substring <span class="tex-font-style-tt">00</span> from the string <span class="tex-font-style-tt">1<span class="tex-font-style-bf">00</span>111</span>, we get <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-353"><span class="MJXp-mn" id="MJXp-Span-354">1</span><span class="MJXp-mo" id="MJXp-Span-355" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mn" id="MJXp-Span-356">2</span><span class="MJXp-mo" id="MJXp-Span-357" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mo" id="MJXp-Span-358" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-359" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-360">4</span><span class="MJXp-mo" id="MJXp-Span-361" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-362" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mo" id="MJXp-Span-363" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-364">2</span></span></span><script type="math/tex" id="MathJax-Element-74">1 \cdot 2 + (-4) = -2</script> points, and the string will be equal to <span class="tex-font-style-tt">1111</span>, removing it entirely we get <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-365"><span class="MJXp-mn" id="MJXp-Span-366">1</span><span class="MJXp-mo" id="MJXp-Span-367" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-mn" id="MJXp-Span-368">4</span><span class="MJXp-mo" id="MJXp-Span-369" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mo" id="MJXp-Span-370" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-371" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-372">4</span><span class="MJXp-mo" id="MJXp-Span-373" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-374" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-375">0</span></span></span><script type="math/tex" id="MathJax-Element-75">1 \cdot 4 + (-4) = 0</script> points. In total, we got <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-376"><span class="MJXp-mo" id="MJXp-Span-377" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-mn" id="MJXp-Span-378">2</span></span></span><script type="math/tex" id="MathJax-Element-76">-2</script> points for <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-379"><span class="MJXp-mn" id="MJXp-Span-380">2</span></span></span><script type="math/tex" id="MathJax-Element-77">2</script> operations.</p></div></div><p> </p></div>
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||
<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">C. Manhattan Subarrays</div><div class="time-limit"><div class="property-title">time limit per test</div>2 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file input-standard"><div class="property-title">input</div>standard input</div><div class="output-file output-standard"><div class="property-title">output</div>standard output</div></div><div><p>Suppose you have two points <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-381"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-382">p</span><span class="MJXp-mo" id="MJXp-Span-383" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mo" id="MJXp-Span-384" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-385"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-386" style="margin-right: 0.05em;">x</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-387" style="vertical-align: -0.4em;">p</span></span><span class="MJXp-mo" id="MJXp-Span-388" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-389"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-390" style="margin-right: 0.05em;">y</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-391" style="vertical-align: -0.4em;">p</span></span><span class="MJXp-mo" id="MJXp-Span-392" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-78">p = (x_p, y_p)</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-393"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-394">q</span><span class="MJXp-mo" id="MJXp-Span-395" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mo" id="MJXp-Span-396" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-397"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-398" style="margin-right: 0.05em;">x</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-399" style="vertical-align: -0.4em;">q</span></span><span class="MJXp-mo" id="MJXp-Span-400" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-401"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-402" style="margin-right: 0.05em;">y</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-403" style="vertical-align: -0.4em;">q</span></span><span class="MJXp-mo" id="MJXp-Span-404" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-79">q = (x_q, y_q)</script>. Let's denote the Manhattan distance between them as <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-405"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-406">d</span><span class="MJXp-mo" id="MJXp-Span-407" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-408">p</span><span class="MJXp-mo" id="MJXp-Span-409" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-410">q</span><span class="MJXp-mo" id="MJXp-Span-411" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-412" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-413"><span class="MJXp-mo" id="MJXp-Span-414" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-msubsup" id="MJXp-Span-415"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-416" style="margin-right: 0.05em;">x</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-417" style="vertical-align: -0.4em;">p</span></span><span class="MJXp-mo" id="MJXp-Span-418" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-419"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-420" style="margin-right: 0.05em;">x</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-421" style="vertical-align: -0.4em;">q</span></span><span class="MJXp-mrow" id="MJXp-Span-422"><span class="MJXp-mo" id="MJXp-Span-423" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-424" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mrow" id="MJXp-Span-425"><span class="MJXp-mo" id="MJXp-Span-426" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-msubsup" id="MJXp-Span-427"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-428" style="margin-right: 0.05em;">y</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-429" style="vertical-align: -0.4em;">p</span></span><span class="MJXp-mo" id="MJXp-Span-430" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-431"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-432" style="margin-right: 0.05em;">y</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-433" style="vertical-align: -0.4em;">q</span></span><span class="MJXp-mrow" id="MJXp-Span-434"><span class="MJXp-mo" id="MJXp-Span-435" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span></span></span><script type="math/tex" id="MathJax-Element-80">d(p, q) = |x_p - x_q| + |y_p - y_q|</script>.</p><p>Let's say that three points <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-436"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-437">p</span></span></span><script type="math/tex" id="MathJax-Element-81">p</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-438"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-439">q</span></span></span><script type="math/tex" id="MathJax-Element-82">q</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-440"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-441">r</span></span></span><script type="math/tex" id="MathJax-Element-83">r</script> form a <span class="tex-font-style-it">bad</span> triple if <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-442"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-443">d</span><span class="MJXp-mo" id="MJXp-Span-444" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-445">p</span><span class="MJXp-mo" id="MJXp-Span-446" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-447">r</span><span class="MJXp-mo" id="MJXp-Span-448" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-449" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-450">d</span><span class="MJXp-mo" id="MJXp-Span-451" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-452">p</span><span class="MJXp-mo" id="MJXp-Span-453" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-454">q</span><span class="MJXp-mo" id="MJXp-Span-455" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-456" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-457">d</span><span class="MJXp-mo" id="MJXp-Span-458" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-459">q</span><span class="MJXp-mo" id="MJXp-Span-460" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-461">r</span><span class="MJXp-mo" id="MJXp-Span-462" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-84">d(p, r) = d(p, q) + d(q, r)</script>.</p><p>Let's say that an array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-463"><span class="MJXp-msubsup" id="MJXp-Span-464"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-465" style="margin-right: 0.05em;">b</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-466" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-467" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-468"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-469" style="margin-right: 0.05em;">b</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-470" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-471" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-472" style="margin-left: 0em; margin-right: 0em;">…</span><span class="MJXp-mo" id="MJXp-Span-473" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-474"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-475" style="margin-right: 0.05em;">b</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-476" style="vertical-align: -0.4em;">m</span></span></span></span><script type="math/tex" id="MathJax-Element-85">b_1, b_2, \dots, b_m</script> is <span class="tex-font-style-it">good</span> if it is impossible to choose three <span class="tex-font-style-bf">distinct</span> indices <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-477"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-478">i</span></span></span><script type="math/tex" id="MathJax-Element-86">i</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-479"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-480">j</span></span></span><script type="math/tex" id="MathJax-Element-87">j</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-481"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-482">k</span></span></span><script type="math/tex" id="MathJax-Element-88">k</script> such that the points <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-483"><span class="MJXp-mo" id="MJXp-Span-484" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-485"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-486" style="margin-right: 0.05em;">b</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-487" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-488" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-489">i</span><span class="MJXp-mo" id="MJXp-Span-490" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-89">(b_i, i)</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-491"><span class="MJXp-mo" id="MJXp-Span-492" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-493"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-494" style="margin-right: 0.05em;">b</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-495" style="vertical-align: -0.4em;">j</span></span><span class="MJXp-mo" id="MJXp-Span-496" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-497">j</span><span class="MJXp-mo" id="MJXp-Span-498" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-90">(b_j, j)</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-499"><span class="MJXp-mo" id="MJXp-Span-500" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-501"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-502" style="margin-right: 0.05em;">b</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-503" style="vertical-align: -0.4em;">k</span></span><span class="MJXp-mo" id="MJXp-Span-504" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-505">k</span><span class="MJXp-mo" id="MJXp-Span-506" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-91">(b_k, k)</script> form a bad triple.</p><p>You are given an array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-507"><span class="MJXp-msubsup" id="MJXp-Span-508"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-509" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-510" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-511" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-512"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-513" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-514" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-515" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-516" style="margin-left: 0em; margin-right: 0em;">…</span><span class="MJXp-mo" id="MJXp-Span-517" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-518"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-519" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-520" style="vertical-align: -0.4em;">n</span></span></span></span><script type="math/tex" id="MathJax-Element-92">a_1, a_2, \dots, a_n</script>. Calculate the number of <span class="tex-font-style-it">good</span> subarrays of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-521"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-522">a</span></span></span><script type="math/tex" id="MathJax-Element-93">a</script>. A subarray of the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-523"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-524">a</span></span></span><script type="math/tex" id="MathJax-Element-94">a</script> is the array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-525"><span class="MJXp-msubsup" id="MJXp-Span-526"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-527" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-528" style="vertical-align: -0.4em;">l</span></span><span class="MJXp-mo" id="MJXp-Span-529" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-530"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-531" style="margin-right: 0.05em;">a</span><span class="MJXp-mrow MJXp-script" id="MJXp-Span-532" style="vertical-align: -0.4em;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-533">l</span><span class="MJXp-mo" id="MJXp-Span-534">+</span><span class="MJXp-mn" id="MJXp-Span-535">1</span></span></span><span class="MJXp-mo" id="MJXp-Span-536" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-537" style="margin-left: 0em; margin-right: 0em;">…</span><span class="MJXp-mo" id="MJXp-Span-538" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-539"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-540" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-541" style="vertical-align: -0.4em;">r</span></span></span></span><script type="math/tex" id="MathJax-Element-95">a_l, a_{l + 1}, \dots, a_r</script> for some <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-542"><span class="MJXp-mn" id="MJXp-Span-543">1</span><span class="MJXp-mo" id="MJXp-Span-544" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-545">l</span><span class="MJXp-mo" id="MJXp-Span-546" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-547">r</span><span class="MJXp-mo" id="MJXp-Span-548" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-549">n</span></span></span><script type="math/tex" id="MathJax-Element-96">1 \le l \le r \le n</script>.</p><p>Note that, according to the definition, subarrays of length <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-550"><span class="MJXp-mn" id="MJXp-Span-551">1</span></span></span><script type="math/tex" id="MathJax-Element-97">1</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-552"><span class="MJXp-mn" id="MJXp-Span-553">2</span></span></span><script type="math/tex" id="MathJax-Element-98">2</script> are <span class="tex-font-style-it">good</span>.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains one integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-554"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-555">t</span></span></span><script type="math/tex" id="MathJax-Element-99">t</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-556"><span class="MJXp-mn" id="MJXp-Span-557">1</span><span class="MJXp-mo" id="MJXp-Span-558" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-559">t</span><span class="MJXp-mo" id="MJXp-Span-560" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-561">5000</span></span></span><script type="math/tex" id="MathJax-Element-100">1 \le t \le 5000</script>) — the number of test cases.</p><p>The first line of each test case contains one integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-562"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-563">n</span></span></span><script type="math/tex" id="MathJax-Element-101">n</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-564"><span class="MJXp-mn" id="MJXp-Span-565">1</span><span class="MJXp-mo" id="MJXp-Span-566" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-567">n</span><span class="MJXp-mo" id="MJXp-Span-568" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-569">2</span><span class="MJXp-mo" id="MJXp-Span-570" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-msubsup" id="MJXp-Span-571"><span class="MJXp-mn" id="MJXp-Span-572" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-573" style="vertical-align: 0.5em;">5</span></span></span></span><script type="math/tex" id="MathJax-Element-102">1 \le n \le 2 \cdot 10^5</script>) — the length of array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-574"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-575">a</span></span></span><script type="math/tex" id="MathJax-Element-103">a</script>.</p><p>The second line of each test case contains <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-576"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-577">n</span></span></span><script type="math/tex" id="MathJax-Element-104">n</script> integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-578"><span class="MJXp-msubsup" id="MJXp-Span-579"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-580" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-581" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-582" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-583"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-584" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-585" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-586" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-587" style="margin-left: 0em; margin-right: 0em;">…</span><span class="MJXp-mo" id="MJXp-Span-588" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-589"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-590" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-591" style="vertical-align: -0.4em;">n</span></span></span></span><script type="math/tex" id="MathJax-Element-105">a_1, a_2, \dots, a_n</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-592"><span class="MJXp-mn" id="MJXp-Span-593">1</span><span class="MJXp-mo" id="MJXp-Span-594" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-595"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-596" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-597" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-598" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-599"><span class="MJXp-mn" id="MJXp-Span-600" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-601" style="vertical-align: 0.5em;">9</span></span></span></span><script type="math/tex" id="MathJax-Element-106">1 \le a_i \le 10^9</script>).</p><p>It's guaranteed that the sum of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-602"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-603">n</span></span></span><script type="math/tex" id="MathJax-Element-107">n</script> doesn't exceed <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-604"><span class="MJXp-mn" id="MJXp-Span-605">2</span><span class="MJXp-mo" id="MJXp-Span-606" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-msubsup" id="MJXp-Span-607"><span class="MJXp-mn" id="MJXp-Span-608" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-609" style="vertical-align: 0.5em;">5</span></span></span></span><script type="math/tex" id="MathJax-Element-108">2 \cdot 10^5</script>.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, print the number of <span class="tex-font-style-it">good</span> subarrays of array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-610"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-611">a</span></span></span><script type="math/tex" id="MathJax-Element-109">a</script>.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id0045100789178148193" id="id0034855463835277845" class="input-output-copier">Copy</div></div><pre id="id0045100789178148193">3
|
||
4
|
||
2 4 1 3
|
||
5
|
||
6 9 1 9 6
|
||
2
|
||
13 37
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id008205050640892653" id="id00646096671878121" class="input-output-copier">Copy</div></div><pre id="id008205050640892653">10
|
||
12
|
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3
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first test case, it can be proven that any subarray of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-612"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-613">a</span></span></span><script type="math/tex" id="MathJax-Element-110">a</script> is <span class="tex-font-style-it">good</span>. For example, subarray <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-614"><span class="MJXp-mo" id="MJXp-Span-615" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-msubsup" id="MJXp-Span-616"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-617" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-618" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-619" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-620"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-621" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-622" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-623" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-624"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-625" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-626" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-627" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-111">[a_2, a_3, a_4]</script> is good since it contains only three elements and: </p><ul><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-628"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-629">d</span><span class="MJXp-mo" id="MJXp-Span-630" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-631" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-632"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-633" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-634" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-635" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-636">2</span><span class="MJXp-mo" id="MJXp-Span-637" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-638" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-639" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-640"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-641" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-642" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-643" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-644">4</span><span class="MJXp-mo" id="MJXp-Span-645" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-646" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-647" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-648"><span class="MJXp-mo" id="MJXp-Span-649" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-650">4</span><span class="MJXp-mo" id="MJXp-Span-651" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-652">3</span><span class="MJXp-mrow" id="MJXp-Span-653"><span class="MJXp-mo" id="MJXp-Span-654" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-655" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mrow" id="MJXp-Span-656"><span class="MJXp-mo" id="MJXp-Span-657" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-658">2</span><span class="MJXp-mo" id="MJXp-Span-659" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-660">4</span><span class="MJXp-mrow" id="MJXp-Span-661"><span class="MJXp-mo" id="MJXp-Span-662" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-663" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-664">3</span></span></span><script type="math/tex" id="MathJax-Element-112">d((a_2, 2), (a_4, 4)) = |4 - 3| + |2 - 4| = 3</script><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-665"><span class="MJXp-mo" id="MJXp-Span-666" style="margin-left: 0.333em; margin-right: 0.333em;"><</span></span></span><script type="math/tex" id="MathJax-Element-113"><</script><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-667"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-668">d</span><span class="MJXp-mo" id="MJXp-Span-669" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-670" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-671"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-672" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-673" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-674" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-675">2</span><span class="MJXp-mo" id="MJXp-Span-676" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-677" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-678" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-679"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-680" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-681" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-682" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-683">3</span><span class="MJXp-mo" id="MJXp-Span-684" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-685" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-686" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-687">d</span><span class="MJXp-mo" id="MJXp-Span-688" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-689" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-690"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-691" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-692" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-693" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-694">3</span><span class="MJXp-mo" id="MJXp-Span-695" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-696" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-697" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-698"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-699" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-700" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-701" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-702">4</span><span class="MJXp-mo" id="MJXp-Span-703" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-704" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-705" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-706">3</span><span class="MJXp-mo" id="MJXp-Span-707" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-708">1</span><span class="MJXp-mo" id="MJXp-Span-709" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-710">2</span><span class="MJXp-mo" id="MJXp-Span-711" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-712">1</span><span class="MJXp-mo" id="MJXp-Span-713" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-714">7</span></span></span><script type="math/tex" id="MathJax-Element-114">d((a_2, 2), (a_3, 3)) + d((a_3, 3), (a_4, 4)) = 3 + 1 + 2 + 1 = 7</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-715"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-716">d</span><span class="MJXp-mo" id="MJXp-Span-717" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-718" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-719"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-720" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-721" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-722" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-723">2</span><span class="MJXp-mo" id="MJXp-Span-724" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-725" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-726" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-727"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-728" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-729" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-730" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-731">3</span><span class="MJXp-mo" id="MJXp-Span-732" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-733" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-115">d((a_2, 2), (a_3, 3))</script><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-734"><span class="MJXp-mo" id="MJXp-Span-735" style="margin-left: 0.333em; margin-right: 0.333em;"><</span></span></span><script type="math/tex" id="MathJax-Element-116"><</script><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-736"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-737">d</span><span class="MJXp-mo" id="MJXp-Span-738" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-739" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-740"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-741" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-742" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-743" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-744">2</span><span class="MJXp-mo" id="MJXp-Span-745" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-746" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-747" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-748"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-749" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-750" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-751" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-752">4</span><span class="MJXp-mo" id="MJXp-Span-753" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-754" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-755" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-756">d</span><span class="MJXp-mo" id="MJXp-Span-757" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-758" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-759"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-760" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-761" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-762" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-763">4</span><span class="MJXp-mo" id="MJXp-Span-764" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-765" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-766" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-767"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-768" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-769" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-770" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-771">3</span><span class="MJXp-mo" id="MJXp-Span-772" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-773" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-117">d((a_2, 2), (a_4, 4)) + d((a_4, 4), (a_3, 3))</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-774"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-775">d</span><span class="MJXp-mo" id="MJXp-Span-776" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-777" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-778"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-779" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-780" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-781" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-782">3</span><span class="MJXp-mo" id="MJXp-Span-783" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-784" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-785" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-786"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-787" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-788" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-789" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-790">4</span><span class="MJXp-mo" id="MJXp-Span-791" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-792" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-118">d((a_3, 3), (a_4, 4))</script><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-793"><span class="MJXp-mo" id="MJXp-Span-794" style="margin-left: 0.333em; margin-right: 0.333em;"><</span></span></span><script type="math/tex" id="MathJax-Element-119"><</script><span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-795"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-796">d</span><span class="MJXp-mo" id="MJXp-Span-797" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-798" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-799"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-800" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-801" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-802" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-803">3</span><span class="MJXp-mo" id="MJXp-Span-804" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-805" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-806" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-807"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-808" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-809" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-810" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-811">2</span><span class="MJXp-mo" id="MJXp-Span-812" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-813" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-814" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-815">d</span><span class="MJXp-mo" id="MJXp-Span-816" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-817" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-818"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-819" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-820" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-821" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-822">2</span><span class="MJXp-mo" id="MJXp-Span-823" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-824" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-825" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-826"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-827" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-828" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-829" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-830">4</span><span class="MJXp-mo" id="MJXp-Span-831" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-832" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-120">d((a_3, 3), (a_2, 2)) + d((a_2, 2), (a_4, 4))</script>; </li></ul><p>In the second test case, for example, subarray <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-833"><span class="MJXp-mo" id="MJXp-Span-834" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-msubsup" id="MJXp-Span-835"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-836" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-837" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-838" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-839"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-840" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-841" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-842" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-843"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-844" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-845" style="vertical-align: -0.4em;">3</span></span><span class="MJXp-mo" id="MJXp-Span-846" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-847"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-848" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-849" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-850" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-121">[a_1, a_2, a_3, a_4]</script> is <span class="tex-font-style-bf">not</span> good, since it contains a <span class="tex-font-style-it">bad</span> triple <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-851"><span class="MJXp-mo" id="MJXp-Span-852" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-853"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-854" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-855" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-856" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-857">1</span><span class="MJXp-mo" id="MJXp-Span-858" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-122">(a_1, 1)</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-859"><span class="MJXp-mo" id="MJXp-Span-860" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-861"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-862" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-863" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-864" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-865">2</span><span class="MJXp-mo" id="MJXp-Span-866" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-123">(a_2, 2)</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-867"><span class="MJXp-mo" id="MJXp-Span-868" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-869"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-870" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-871" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-872" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-873">4</span><span class="MJXp-mo" id="MJXp-Span-874" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-124">(a_4, 4)</script>: </p><ul><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-875"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-876">d</span><span class="MJXp-mo" id="MJXp-Span-877" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-878" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-879"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-880" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-881" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-882" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-883">1</span><span class="MJXp-mo" id="MJXp-Span-884" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-885" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-886" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-887"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-888" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-889" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-890" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-891">4</span><span class="MJXp-mo" id="MJXp-Span-892" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-893" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-894" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-895"><span class="MJXp-mo" id="MJXp-Span-896" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-897">6</span><span class="MJXp-mo" id="MJXp-Span-898" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-899">9</span><span class="MJXp-mrow" id="MJXp-Span-900"><span class="MJXp-mo" id="MJXp-Span-901" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-902" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mrow" id="MJXp-Span-903"><span class="MJXp-mo" id="MJXp-Span-904" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-905">1</span><span class="MJXp-mo" id="MJXp-Span-906" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-907">4</span><span class="MJXp-mrow" id="MJXp-Span-908"><span class="MJXp-mo" id="MJXp-Span-909" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-910" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-911">6</span></span></span><script type="math/tex" id="MathJax-Element-125">d((a_1, 1), (a_4, 4)) = |6 - 9| + |1 - 4| = 6</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-912"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-913">d</span><span class="MJXp-mo" id="MJXp-Span-914" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-915" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-916"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-917" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-918" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-919" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-920">1</span><span class="MJXp-mo" id="MJXp-Span-921" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-922" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-923" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-924"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-925" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-926" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-927" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-928">2</span><span class="MJXp-mo" id="MJXp-Span-929" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-930" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-931" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-932"><span class="MJXp-mo" id="MJXp-Span-933" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-934">6</span><span class="MJXp-mo" id="MJXp-Span-935" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-936">9</span><span class="MJXp-mrow" id="MJXp-Span-937"><span class="MJXp-mo" id="MJXp-Span-938" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-939" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mrow" id="MJXp-Span-940"><span class="MJXp-mo" id="MJXp-Span-941" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-942">1</span><span class="MJXp-mo" id="MJXp-Span-943" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-944">2</span><span class="MJXp-mrow" id="MJXp-Span-945"><span class="MJXp-mo" id="MJXp-Span-946" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-947" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-948">4</span></span></span><script type="math/tex" id="MathJax-Element-126">d((a_1, 1), (a_2, 2)) = |6 - 9| + |1 - 2| = 4</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-949"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-950">d</span><span class="MJXp-mo" id="MJXp-Span-951" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-952" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-953"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-954" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-955" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-956" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-957">2</span><span class="MJXp-mo" id="MJXp-Span-958" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-959" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-960" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-961"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-962" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-963" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-964" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-965">4</span><span class="MJXp-mo" id="MJXp-Span-966" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-967" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-968" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mrow" id="MJXp-Span-969"><span class="MJXp-mo" id="MJXp-Span-970" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-971">9</span><span class="MJXp-mo" id="MJXp-Span-972" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-973">9</span><span class="MJXp-mrow" id="MJXp-Span-974"><span class="MJXp-mo" id="MJXp-Span-975" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-976" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mrow" id="MJXp-Span-977"><span class="MJXp-mo" id="MJXp-Span-978" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mn" id="MJXp-Span-979">2</span><span class="MJXp-mo" id="MJXp-Span-980" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-981">4</span><span class="MJXp-mrow" id="MJXp-Span-982"><span class="MJXp-mo" id="MJXp-Span-983" style="margin-left: 0.167em; margin-right: 0.167em;">|</span></span><span class="MJXp-mo" id="MJXp-Span-984" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-985">2</span></span></span><script type="math/tex" id="MathJax-Element-127">d((a_2, 2), (a_4, 4)) = |9 - 9| + |2 - 4| = 2</script>; </li></ul><p>So, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-986"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-987">d</span><span class="MJXp-mo" id="MJXp-Span-988" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-989" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-990"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-991" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-992" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-993" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-994">1</span><span class="MJXp-mo" id="MJXp-Span-995" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-996" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-997" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-998"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-999" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1000" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-1001" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1002">4</span><span class="MJXp-mo" id="MJXp-Span-1003" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1004" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1005" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1006">d</span><span class="MJXp-mo" id="MJXp-Span-1007" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-1008" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-1009"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1010" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1011" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1012" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1013">1</span><span class="MJXp-mo" id="MJXp-Span-1014" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1015" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-1016" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-1017"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1018" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1019" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1020" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1021">2</span><span class="MJXp-mo" id="MJXp-Span-1022" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1023" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1024" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1025">d</span><span class="MJXp-mo" id="MJXp-Span-1026" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mo" id="MJXp-Span-1027" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-1028"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1029" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1030" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1031" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1032">2</span><span class="MJXp-mo" id="MJXp-Span-1033" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1034" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-1035" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-msubsup" id="MJXp-Span-1036"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1037" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1038" style="vertical-align: -0.4em;">4</span></span><span class="MJXp-mo" id="MJXp-Span-1039" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1040">4</span><span class="MJXp-mo" id="MJXp-Span-1041" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mo" id="MJXp-Span-1042" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-128">d((a_1, 1), (a_4, 4)) = d((a_1, 1), (a_2, 2)) + d((a_2, 2), (a_4, 4))</script>.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">D. Excellent Arrays</div><div class="time-limit"><div class="property-title">time limit per test</div>2 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file input-standard"><div class="property-title">input</div>standard input</div><div class="output-file output-standard"><div class="property-title">output</div>standard output</div></div><div><p>Let's call an integer array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1043"><span class="MJXp-msubsup" id="MJXp-Span-1044"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1045" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1046" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1047" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-1048"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1049" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1050" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1051" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-1052" style="margin-left: 0em; margin-right: 0em;">…</span><span class="MJXp-mo" id="MJXp-Span-1053" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-1054"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1055" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1056" style="vertical-align: -0.4em;">n</span></span></span></span><script type="math/tex" id="MathJax-Element-129">a_1, a_2, \dots, a_n</script><span class="tex-font-style-it">good</span> if <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1057"><span class="MJXp-msubsup" id="MJXp-Span-1058"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1059" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1060" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-1061" style="margin-left: 0.333em; margin-right: 0.333em;">≠</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1062">i</span></span></span><script type="math/tex" id="MathJax-Element-130">a_i \neq i</script> for each <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1063"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1064">i</span></span></span><script type="math/tex" id="MathJax-Element-131">i</script>.</p><p>Let <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1065"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1066">F</span><span class="MJXp-mo" id="MJXp-Span-1067" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1068">a</span><span class="MJXp-mo" id="MJXp-Span-1069" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-132">F(a)</script> be the number of pairs <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1070"><span class="MJXp-mo" id="MJXp-Span-1071" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1072">i</span><span class="MJXp-mo" id="MJXp-Span-1073" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1074">j</span><span class="MJXp-mo" id="MJXp-Span-1075" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-133">(i, j)</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1076"><span class="MJXp-mn" id="MJXp-Span-1077">1</span><span class="MJXp-mo" id="MJXp-Span-1078" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1079">i</span><span class="MJXp-mo" id="MJXp-Span-1080" style="margin-left: 0.333em; margin-right: 0.333em;"><</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1081">j</span><span class="MJXp-mo" id="MJXp-Span-1082" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1083">n</span></span></span><script type="math/tex" id="MathJax-Element-134">1 \le i < j \le n</script>) such that <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1084"><span class="MJXp-msubsup" id="MJXp-Span-1085"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1086" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1087" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-1088" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-msubsup" id="MJXp-Span-1089"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1090" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1091" style="vertical-align: -0.4em;">j</span></span><span class="MJXp-mo" id="MJXp-Span-1092" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1093">i</span><span class="MJXp-mo" id="MJXp-Span-1094" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1095">j</span></span></span><script type="math/tex" id="MathJax-Element-135">a_i + a_j = i + j</script>.</p><p>Let's say that an array <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1096"><span class="MJXp-msubsup" id="MJXp-Span-1097"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1098" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1099" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1100" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-1101"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1102" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1103" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1104" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-1105" style="margin-left: 0em; margin-right: 0em;">…</span><span class="MJXp-mo" id="MJXp-Span-1106" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-1107"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1108" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1109" style="vertical-align: -0.4em;">n</span></span></span></span><script type="math/tex" id="MathJax-Element-136">a_1, a_2, \dots, a_n</script> is <span class="tex-font-style-it">excellent</span> if: </p><ul><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1110"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1111">a</span></span></span><script type="math/tex" id="MathJax-Element-137">a</script> is <span class="tex-font-style-it">good</span>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1112"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1113">l</span><span class="MJXp-mo" id="MJXp-Span-1114" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-1115"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1116" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1117" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-1118" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1119">r</span></span></span><script type="math/tex" id="MathJax-Element-138">l \le a_i \le r</script> for each <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1120"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1121">i</span></span></span><script type="math/tex" id="MathJax-Element-139">i</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1122"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1123">F</span><span class="MJXp-mo" id="MJXp-Span-1124" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1125">a</span><span class="MJXp-mo" id="MJXp-Span-1126" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-140">F(a)</script> is the maximum possible among all <span class="tex-font-style-it">good</span> arrays of size <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1127"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1128">n</span></span></span><script type="math/tex" id="MathJax-Element-141">n</script>. </li></ul><p>Given <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1129"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1130">n</span></span></span><script type="math/tex" id="MathJax-Element-142">n</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1131"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1132">l</span></span></span><script type="math/tex" id="MathJax-Element-143">l</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1133"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1134">r</span></span></span><script type="math/tex" id="MathJax-Element-144">r</script>, calculate the number of <span class="tex-font-style-it">excellent</span> arrays modulo <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1135"><span class="MJXp-msubsup" id="MJXp-Span-1136"><span class="MJXp-mn" id="MJXp-Span-1137" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1138" style="vertical-align: 0.5em;">9</span></span><span class="MJXp-mo" id="MJXp-Span-1139" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-1140">7</span></span></span><script type="math/tex" id="MathJax-Element-145">10^9 + 7</script>.</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains a single integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1141"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1142">t</span></span></span><script type="math/tex" id="MathJax-Element-146">t</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1143"><span class="MJXp-mn" id="MJXp-Span-1144">1</span><span class="MJXp-mo" id="MJXp-Span-1145" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1146">t</span><span class="MJXp-mo" id="MJXp-Span-1147" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-1148">1000</span></span></span><script type="math/tex" id="MathJax-Element-147">1 \le t \le 1000</script>) — the number of test cases. </p><p>The first and only line of each test case contains three integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1149"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1150">n</span></span></span><script type="math/tex" id="MathJax-Element-148">n</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1151"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1152">l</span></span></span><script type="math/tex" id="MathJax-Element-149">l</script>, and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1153"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1154">r</span></span></span><script type="math/tex" id="MathJax-Element-150">r</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1155"><span class="MJXp-mn" id="MJXp-Span-1156">2</span><span class="MJXp-mo" id="MJXp-Span-1157" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1158">n</span><span class="MJXp-mo" id="MJXp-Span-1159" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-1160">2</span><span class="MJXp-mo" id="MJXp-Span-1161" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-msubsup" id="MJXp-Span-1162"><span class="MJXp-mn" id="MJXp-Span-1163" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1164" style="vertical-align: 0.5em;">5</span></span></span></span><script type="math/tex" id="MathJax-Element-151">2 \le n \le 2 \cdot 10^5</script>; <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1165"><span class="MJXp-mo" id="MJXp-Span-1166" style="margin-left: 0em; margin-right: 0.111em;">−</span><span class="MJXp-msubsup" id="MJXp-Span-1167"><span class="MJXp-mn" id="MJXp-Span-1168" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1169" style="vertical-align: 0.5em;">9</span></span><span class="MJXp-mo" id="MJXp-Span-1170" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1171">l</span><span class="MJXp-mo" id="MJXp-Span-1172" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-1173">1</span></span></span><script type="math/tex" id="MathJax-Element-152">-10^9 \le l \le 1</script>; <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1174"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1175">n</span><span class="MJXp-mo" id="MJXp-Span-1176" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1177">r</span><span class="MJXp-mo" id="MJXp-Span-1178" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-1179"><span class="MJXp-mn" id="MJXp-Span-1180" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1181" style="vertical-align: 0.5em;">9</span></span></span></span><script type="math/tex" id="MathJax-Element-153">n \le r \le 10^9</script>).</p><p>It's guaranteed that the sum of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1182"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1183">n</span></span></span><script type="math/tex" id="MathJax-Element-154">n</script> doesn't exceed <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1184"><span class="MJXp-mn" id="MJXp-Span-1185">2</span><span class="MJXp-mo" id="MJXp-Span-1186" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-msubsup" id="MJXp-Span-1187"><span class="MJXp-mn" id="MJXp-Span-1188" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1189" style="vertical-align: 0.5em;">5</span></span></span></span><script type="math/tex" id="MathJax-Element-155">2 \cdot 10^5</script>.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each test case, print the number of excellent arrays modulo <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1190"><span class="MJXp-msubsup" id="MJXp-Span-1191"><span class="MJXp-mn" id="MJXp-Span-1192" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1193" style="vertical-align: 0.5em;">9</span></span><span class="MJXp-mo" id="MJXp-Span-1194" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-1195">7</span></span></span><script type="math/tex" id="MathJax-Element-156">10^9 + 7</script>.</p></div><div class="sample-tests"><div class="section-title">Example</div><div class="sample-test"><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id0023043713167808821" id="id004407875898011011" class="input-output-copier">Copy</div></div><pre id="id0023043713167808821">4
|
||
3 0 3
|
||
4 -3 5
|
||
42 -33 55
|
||
69 -42 146
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id004542568812571498" id="id008244354808710753" class="input-output-copier">Copy</div></div><pre id="id004542568812571498">4
|
||
10
|
||
143922563
|
||
698570404
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||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first test case, it can be proven that the maximum <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1196"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1197">F</span><span class="MJXp-mo" id="MJXp-Span-1198" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1199">a</span><span class="MJXp-mo" id="MJXp-Span-1200" style="margin-left: 0em; margin-right: 0em;">)</span></span></span><script type="math/tex" id="MathJax-Element-157">F(a)</script> among all <span class="tex-font-style-it">good</span> arrays <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1201"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1202">a</span></span></span><script type="math/tex" id="MathJax-Element-158">a</script> is equal to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1203"><span class="MJXp-mn" id="MJXp-Span-1204">2</span></span></span><script type="math/tex" id="MathJax-Element-159">2</script>. The excellent arrays are: </p><ol><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1205"><span class="MJXp-mo" id="MJXp-Span-1206" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-1207">2</span><span class="MJXp-mo" id="MJXp-Span-1208" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1209">1</span><span class="MJXp-mo" id="MJXp-Span-1210" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1211">2</span><span class="MJXp-mo" id="MJXp-Span-1212" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-160">[2, 1, 2]</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1213"><span class="MJXp-mo" id="MJXp-Span-1214" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-1215">0</span><span class="MJXp-mo" id="MJXp-Span-1216" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1217">3</span><span class="MJXp-mo" id="MJXp-Span-1218" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1219">2</span><span class="MJXp-mo" id="MJXp-Span-1220" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-161">[0, 3, 2]</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1221"><span class="MJXp-mo" id="MJXp-Span-1222" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-1223">2</span><span class="MJXp-mo" id="MJXp-Span-1224" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1225">3</span><span class="MJXp-mo" id="MJXp-Span-1226" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1227">2</span><span class="MJXp-mo" id="MJXp-Span-1228" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-162">[2, 3, 2]</script>; </li><li> <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1229"><span class="MJXp-mo" id="MJXp-Span-1230" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-1231">3</span><span class="MJXp-mo" id="MJXp-Span-1232" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1233">0</span><span class="MJXp-mo" id="MJXp-Span-1234" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mn" id="MJXp-Span-1235">1</span><span class="MJXp-mo" id="MJXp-Span-1236" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-163">[3, 0, 1]</script>. </li></ol></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">E. Stringforces</div><div class="time-limit"><div class="property-title">time limit per test</div>3 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file input-standard"><div class="property-title">input</div>standard input</div><div class="output-file output-standard"><div class="property-title">output</div>standard output</div></div><div><p>You are given a string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1237"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1238">s</span></span></span><script type="math/tex" id="MathJax-Element-164">s</script> of length <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1239"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1240">n</span></span></span><script type="math/tex" id="MathJax-Element-165">n</script>. Each character is either one of the first <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1241"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1242">k</span></span></span><script type="math/tex" id="MathJax-Element-166">k</script> lowercase Latin letters or a question mark.</p><p>You are asked to replace every question mark with one of the first <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1243"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1244">k</span></span></span><script type="math/tex" id="MathJax-Element-167">k</script> lowercase Latin letters in such a way that the following value is maximized.</p><p>Let <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1245"><span class="MJXp-msubsup" id="MJXp-Span-1246"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1247" style="margin-right: 0.05em;">f</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1248" style="vertical-align: -0.4em;">i</span></span></span></span><script type="math/tex" id="MathJax-Element-168">f_i</script> be the maximum length substring of string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1249"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1250">s</span></span></span><script type="math/tex" id="MathJax-Element-169">s</script>, which consists entirely of the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1251"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1252">i</span></span></span><script type="math/tex" id="MathJax-Element-170">i</script>-th Latin letter. A substring of a string is a contiguous subsequence of that string. If the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1253"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1254">i</span></span></span><script type="math/tex" id="MathJax-Element-171">i</script>-th letter doesn't appear in a string, then <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1255"><span class="MJXp-msubsup" id="MJXp-Span-1256"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1257" style="margin-right: 0.05em;">f</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1258" style="vertical-align: -0.4em;">i</span></span></span></span><script type="math/tex" id="MathJax-Element-172">f_i</script> is equal to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1259"><span class="MJXp-mn" id="MJXp-Span-1260">0</span></span></span><script type="math/tex" id="MathJax-Element-173">0</script>.</p><p>The value of a string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1261"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1262">s</span></span></span><script type="math/tex" id="MathJax-Element-174">s</script> is the minimum value among <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1263"><span class="MJXp-msubsup" id="MJXp-Span-1264"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1265" style="margin-right: 0.05em;">f</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1266" style="vertical-align: -0.4em;">i</span></span></span></span><script type="math/tex" id="MathJax-Element-175">f_i</script> for all <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1267"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1268">i</span></span></span><script type="math/tex" id="MathJax-Element-176">i</script> from <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1269"><span class="MJXp-mn" id="MJXp-Span-1270">1</span></span></span><script type="math/tex" id="MathJax-Element-177">1</script> to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1271"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1272">k</span></span></span><script type="math/tex" id="MathJax-Element-178">k</script>.</p><p>What is the maximum value the string can have?</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains two integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1273"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1274">n</span></span></span><script type="math/tex" id="MathJax-Element-179">n</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1275"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1276">k</span></span></span><script type="math/tex" id="MathJax-Element-180">k</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1277"><span class="MJXp-mn" id="MJXp-Span-1278">1</span><span class="MJXp-mo" id="MJXp-Span-1279" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1280">n</span><span class="MJXp-mo" id="MJXp-Span-1281" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-1282">2</span><span class="MJXp-mo" id="MJXp-Span-1283" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-msubsup" id="MJXp-Span-1284"><span class="MJXp-mn" id="MJXp-Span-1285" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1286" style="vertical-align: 0.5em;">5</span></span></span></span><script type="math/tex" id="MathJax-Element-181">1 \le n \le 2 \cdot 10^5</script>; <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1287"><span class="MJXp-mn" id="MJXp-Span-1288">1</span><span class="MJXp-mo" id="MJXp-Span-1289" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1290">k</span><span class="MJXp-mo" id="MJXp-Span-1291" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-1292">17</span></span></span><script type="math/tex" id="MathJax-Element-182">1 \le k \le 17</script>) — the length of the string and the number of first Latin letters used.</p><p>The second line contains a string <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1293"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1294">s</span></span></span><script type="math/tex" id="MathJax-Element-183">s</script>, consisting of <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1295"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1296">n</span></span></span><script type="math/tex" id="MathJax-Element-184">n</script> characters. Each character is either one of the first <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1297"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1298">k</span></span></span><script type="math/tex" id="MathJax-Element-185">k</script> lowercase Latin letters or a question mark.</p></div><div class="output-specification"><div class="section-title">Output</div><p>Print a single integer — the maximum value of the string after every question mark is replaced with one of the first <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1299"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1300">k</span></span></span><script type="math/tex" id="MathJax-Element-186">k</script> lowercase Latin letters.</p></div><div class="sample-tests"><div class="section-title">Examples</div><div class="sample-test"><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id0010278017840988773" id="id0036697665286741366" class="input-output-copier">Copy</div></div><pre id="id0010278017840988773">10 2
|
||
a??ab????b
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id0012852910736247602" id="id0023007378841179338" class="input-output-copier">Copy</div></div><pre id="id0012852910736247602">4
|
||
</pre></div><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id008360522336769922" id="id006567512747417346" class="input-output-copier">Copy</div></div><pre id="id008360522336769922">9 4
|
||
?????????
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id003666838310571824" id="id007490701155756769" class="input-output-copier">Copy</div></div><pre id="id003666838310571824">2
|
||
</pre></div><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id004808096454233227" id="id0029136739482146945" class="input-output-copier">Copy</div></div><pre id="id004808096454233227">2 3
|
||
??
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id0020261251752636744" id="id008323933857992657" class="input-output-copier">Copy</div></div><pre id="id0020261251752636744">0
|
||
</pre></div><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id006719335306102051" id="id007729559272990804" class="input-output-copier">Copy</div></div><pre id="id006719335306102051">15 3
|
||
??b?babbc??b?aa
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id0011681461962729234" id="id005737182969104904" class="input-output-copier">Copy</div></div><pre id="id0011681461962729234">3
|
||
</pre></div><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id0049861046124584174" id="id00027302885836351765" class="input-output-copier">Copy</div></div><pre id="id0049861046124584174">4 4
|
||
cabd
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id007620779631950757" id="id00005901348145114671" class="input-output-copier">Copy</div></div><pre id="id007620779631950757">1
|
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</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>In the first example the question marks can be replaced in the following way: "<span class="tex-font-style-tt">a<span class="tex-font-style-underline">aa</span>ab<span class="tex-font-style-underline">abbb</span>b</span>". <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1301"><span class="MJXp-msubsup" id="MJXp-Span-1302"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1303" style="margin-right: 0.05em;">f</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1304" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1305" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1306">4</span></span></span><script type="math/tex" id="MathJax-Element-187">f_1 = 4</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1307"><span class="MJXp-msubsup" id="MJXp-Span-1308"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1309" style="margin-right: 0.05em;">f</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1310" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1311" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1312">4</span></span></span><script type="math/tex" id="MathJax-Element-188">f_2 = 4</script>, thus the answer is <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1313"><span class="MJXp-mn" id="MJXp-Span-1314">4</span></span></span><script type="math/tex" id="MathJax-Element-189">4</script>. Replacing it like this is also possible: "<span class="tex-font-style-tt">a<span class="tex-font-style-underline">aa</span>ab<span class="tex-font-style-underline">bbbb</span>b</span>". That way <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1315"><span class="MJXp-msubsup" id="MJXp-Span-1316"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1317" style="margin-right: 0.05em;">f</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1318" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1319" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1320">4</span></span></span><script type="math/tex" id="MathJax-Element-190">f_1 = 4</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1321"><span class="MJXp-msubsup" id="MJXp-Span-1322"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1323" style="margin-right: 0.05em;">f</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1324" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1325" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-1326">6</span></span></span><script type="math/tex" id="MathJax-Element-191">f_2 = 6</script>, however, the minimum of them is still <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1327"><span class="MJXp-mn" id="MJXp-Span-1328">4</span></span></span><script type="math/tex" id="MathJax-Element-192">4</script>.</p><p>In the second example one of the possible strings is "<span class="tex-font-style-tt"><span class="tex-font-style-underline">aabbccdda</span></span>".</p><p>In the third example at least one letter won't appear in the string, thus, the minimum of values <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1329"><span class="MJXp-msubsup" id="MJXp-Span-1330"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1331" style="margin-right: 0.05em;">f</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1332" style="vertical-align: -0.4em;">i</span></span></span></span><script type="math/tex" id="MathJax-Element-193">f_i</script> is always <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1333"><span class="MJXp-mn" id="MJXp-Span-1334">0</span></span></span><script type="math/tex" id="MathJax-Element-194">0</script>.</p></div></div><p> </p></div>
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<div class="ttypography"><div class="problem-statement"><div class="header"><div class="title">F. Jumping Around</div><div class="time-limit"><div class="property-title">time limit per test</div>5 seconds</div><div class="memory-limit"><div class="property-title">memory limit per test</div>256 megabytes</div><div class="input-file input-standard"><div class="property-title">input</div>standard input</div><div class="output-file output-standard"><div class="property-title">output</div>standard output</div></div><div><p>There is an infinite pond that can be represented with a number line. There are <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1335"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1336">n</span></span></span><script type="math/tex" id="MathJax-Element-195">n</script> rocks in the pond, numbered from <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1337"><span class="MJXp-mn" id="MJXp-Span-1338">1</span></span></span><script type="math/tex" id="MathJax-Element-196">1</script> to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1339"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1340">n</span></span></span><script type="math/tex" id="MathJax-Element-197">n</script>. The <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1341"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1342">i</span></span></span><script type="math/tex" id="MathJax-Element-198">i</script>-th rock is located at an integer coordinate <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1343"><span class="MJXp-msubsup" id="MJXp-Span-1344"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1345" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1346" style="vertical-align: -0.4em;">i</span></span></span></span><script type="math/tex" id="MathJax-Element-199">a_i</script>. The coordinates of the rocks are pairwise distinct. The rocks are numbered in the increasing order of the coordinate, so <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1347"><span class="MJXp-msubsup" id="MJXp-Span-1348"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1349" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1350" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1351" style="margin-left: 0.333em; margin-right: 0.333em;"><</span><span class="MJXp-msubsup" id="MJXp-Span-1352"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1353" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1354" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1355" style="margin-left: 0.333em; margin-right: 0.333em;"><</span><span class="MJXp-mo" id="MJXp-Span-1356" style="margin-left: 0em; margin-right: 0em;">⋯</span><span class="MJXp-mo" id="MJXp-Span-1357" style="margin-left: 0.333em; margin-right: 0.333em;"><</span><span class="MJXp-msubsup" id="MJXp-Span-1358"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1359" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1360" style="vertical-align: -0.4em;">n</span></span></span></span><script type="math/tex" id="MathJax-Element-200">a_1 < a_2 < \dots < a_n</script>.</p><p>A robot frog sits on the rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1361"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1362">s</span></span></span><script type="math/tex" id="MathJax-Element-201">s</script>. The frog is programmable. It has a base jumping distance parameter <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1363"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1364">d</span></span></span><script type="math/tex" id="MathJax-Element-202">d</script>. There also is a setting for the jumping distance range. If the jumping distance range is set to some integer <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1365"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1366">k</span></span></span><script type="math/tex" id="MathJax-Element-203">k</script>, then the frog can jump from some rock to any rock at a distance from <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1367"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1368">d</span><span class="MJXp-mo" id="MJXp-Span-1369" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1370">k</span></span></span><script type="math/tex" id="MathJax-Element-204">d - k</script> to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1371"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1372">d</span><span class="MJXp-mo" id="MJXp-Span-1373" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1374">k</span></span></span><script type="math/tex" id="MathJax-Element-205">d + k</script> inclusive in any direction. The distance between two rocks is an absolute difference between their coordinates.</p><p>You are assigned a task to implement a feature for the frog. Given two integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1375"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1376">i</span></span></span><script type="math/tex" id="MathJax-Element-206">i</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1377"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1378">k</span></span></span><script type="math/tex" id="MathJax-Element-207">k</script> determine if the frog can reach a rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1379"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1380">i</span></span></span><script type="math/tex" id="MathJax-Element-208">i</script> from a rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1381"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1382">s</span></span></span><script type="math/tex" id="MathJax-Element-209">s</script> performing a sequence of jumps with the jumping distance range set to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1383"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1384">k</span></span></span><script type="math/tex" id="MathJax-Element-210">k</script>. The sequence can be arbitrarily long or empty.</p><p>You will be given <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1385"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1386">q</span></span></span><script type="math/tex" id="MathJax-Element-211">q</script> testcases for that feature, the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1387"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1388">j</span></span></span><script type="math/tex" id="MathJax-Element-212">j</script>-th testcase consists of two integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1389"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1390">i</span></span></span><script type="math/tex" id="MathJax-Element-213">i</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1391"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1392">k</span></span></span><script type="math/tex" id="MathJax-Element-214">k</script>. Print "<span class="tex-font-style-tt">Yes</span>" if the <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1393"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1394">i</span></span></span><script type="math/tex" id="MathJax-Element-215">i</script>-th rock is reachable and "<span class="tex-font-style-tt">No</span>" otherwise.</p><p>You can output "<span class="tex-font-style-tt">YES</span>" and "<span class="tex-font-style-tt">NO</span>" in any case (for example, strings "<span class="tex-font-style-tt">yEs</span>", "<span class="tex-font-style-tt">yes</span>", "<span class="tex-font-style-tt">Yes</span>" and '<span class="tex-font-style-tt">YES</span>"' will be recognized as a positive answer).</p></div><div class="input-specification"><div class="section-title">Input</div><p>The first line contains four integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1395"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1396">n</span></span></span><script type="math/tex" id="MathJax-Element-216">n</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1397"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1398">q</span></span></span><script type="math/tex" id="MathJax-Element-217">q</script>, <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1399"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1400">s</span></span></span><script type="math/tex" id="MathJax-Element-218">s</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1401"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1402">d</span></span></span><script type="math/tex" id="MathJax-Element-219">d</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1403"><span class="MJXp-mn" id="MJXp-Span-1404">1</span><span class="MJXp-mo" id="MJXp-Span-1405" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1406">n</span><span class="MJXp-mo" id="MJXp-Span-1407" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1408">q</span><span class="MJXp-mo" id="MJXp-Span-1409" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mn" id="MJXp-Span-1410">2</span><span class="MJXp-mo" id="MJXp-Span-1411" style="margin-left: 0.267em; margin-right: 0.267em;">⋅</span><span class="MJXp-msubsup" id="MJXp-Span-1412"><span class="MJXp-mn" id="MJXp-Span-1413" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1414" style="vertical-align: 0.5em;">5</span></span></span></span><script type="math/tex" id="MathJax-Element-220">1 \le n, q \le 2 \cdot 10^5</script>; <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1415"><span class="MJXp-mn" id="MJXp-Span-1416">1</span><span class="MJXp-mo" id="MJXp-Span-1417" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1418">s</span><span class="MJXp-mo" id="MJXp-Span-1419" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1420">n</span></span></span><script type="math/tex" id="MathJax-Element-221">1 \le s \le n</script>; <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1421"><span class="MJXp-mn" id="MJXp-Span-1422">1</span><span class="MJXp-mo" id="MJXp-Span-1423" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1424">d</span><span class="MJXp-mo" id="MJXp-Span-1425" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-1426"><span class="MJXp-mn" id="MJXp-Span-1427" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1428" style="vertical-align: 0.5em;">6</span></span></span></span><script type="math/tex" id="MathJax-Element-222">1 \le d \le 10^6</script>) — the number of rocks, the number of testcases, the starting rock and the base jumping distance parameter.</p><p>The second line contains <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1429"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1430">n</span></span></span><script type="math/tex" id="MathJax-Element-223">n</script> integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1431"><span class="MJXp-msubsup" id="MJXp-Span-1432"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1433" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1434" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1435" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-1436"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1437" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1438" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1439" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-mo" id="MJXp-Span-1440" style="margin-left: 0em; margin-right: 0em;">…</span><span class="MJXp-mo" id="MJXp-Span-1441" style="margin-left: 0em; margin-right: 0.222em;">,</span><span class="MJXp-msubsup" id="MJXp-Span-1442"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1443" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1444" style="vertical-align: -0.4em;">n</span></span></span></span><script type="math/tex" id="MathJax-Element-224">a_1, a_2, \dots, a_n</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1445"><span class="MJXp-mn" id="MJXp-Span-1446">1</span><span class="MJXp-mo" id="MJXp-Span-1447" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-1448"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1449" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1450" style="vertical-align: -0.4em;">i</span></span><span class="MJXp-mo" id="MJXp-Span-1451" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-1452"><span class="MJXp-mn" id="MJXp-Span-1453" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1454" style="vertical-align: 0.5em;">6</span></span></span></span><script type="math/tex" id="MathJax-Element-225">1 \le a_i \le 10^6</script>) — the coordinates of the rocks. The coordinates of the rocks are pairwise distinct. The rocks are numbered in the increasing order of distance from the land, so <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1455"><span class="MJXp-msubsup" id="MJXp-Span-1456"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1457" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1458" style="vertical-align: -0.4em;">1</span></span><span class="MJXp-mo" id="MJXp-Span-1459" style="margin-left: 0.333em; margin-right: 0.333em;"><</span><span class="MJXp-msubsup" id="MJXp-Span-1460"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1461" style="margin-right: 0.05em;">a</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1462" style="vertical-align: -0.4em;">2</span></span><span class="MJXp-mo" id="MJXp-Span-1463" style="margin-left: 0.333em; margin-right: 0.333em;"><</span><span class="MJXp-mo" id="MJXp-Span-1464" style="margin-left: 0em; margin-right: 0em;">⋯</span><span class="MJXp-mo" id="MJXp-Span-1465" style="margin-left: 0.333em; margin-right: 0.333em;"><</span><span class="MJXp-msubsup" id="MJXp-Span-1466"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1467" style="margin-right: 0.05em;">a</span><span class="MJXp-mi MJXp-italic MJXp-script" id="MJXp-Span-1468" style="vertical-align: -0.4em;">n</span></span></span></span><script type="math/tex" id="MathJax-Element-226">a_1 < a_2 < \dots < a_n</script>.</p><p>Each of the next <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1469"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1470">q</span></span></span><script type="math/tex" id="MathJax-Element-227">q</script> lines contains two integers <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1471"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1472">i</span></span></span><script type="math/tex" id="MathJax-Element-228">i</script> and <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1473"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1474">k</span></span></span><script type="math/tex" id="MathJax-Element-229">k</script> (<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1475"><span class="MJXp-mn" id="MJXp-Span-1476">1</span><span class="MJXp-mo" id="MJXp-Span-1477" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1478">i</span><span class="MJXp-mo" id="MJXp-Span-1479" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1480">n</span></span></span><script type="math/tex" id="MathJax-Element-230">1 \le i \le n</script>; <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1481"><span class="MJXp-mn" id="MJXp-Span-1482">1</span><span class="MJXp-mo" id="MJXp-Span-1483" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1484">k</span><span class="MJXp-mo" id="MJXp-Span-1485" style="margin-left: 0.333em; margin-right: 0.333em;">≤</span><span class="MJXp-msubsup" id="MJXp-Span-1486"><span class="MJXp-mn" id="MJXp-Span-1487" style="margin-right: 0.05em;">10</span><span class="MJXp-mn MJXp-script" id="MJXp-Span-1488" style="vertical-align: 0.5em;">6</span></span></span></span><script type="math/tex" id="MathJax-Element-231">1 \le k \le 10^6</script>) — the parameters to the testcase.</p></div><div class="output-specification"><div class="section-title">Output</div><p>For each of the testcases print an answer. If there is a sequence of jumps from a rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1489"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1490">s</span></span></span><script type="math/tex" id="MathJax-Element-232">s</script> to a rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1491"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1492">i</span></span></span><script type="math/tex" id="MathJax-Element-233">i</script> with the jumping distance range set to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1493"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-1494">k</span></span></span><script type="math/tex" id="MathJax-Element-234">k</script>, then print "<span class="tex-font-style-tt">Yes</span>". Otherwise, print "<span class="tex-font-style-tt">No</span>".</p></div><div class="sample-tests"><div class="section-title">Examples</div><div class="sample-test"><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id009457268175320975" id="id0009232387181386903" class="input-output-copier">Copy</div></div><pre id="id009457268175320975">7 4 4 5
|
||
1 5 10 13 20 22 28
|
||
4 1
|
||
7 2
|
||
7 3
|
||
3 2
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id006703894185413486" id="id009325666133454178" class="input-output-copier">Copy</div></div><pre id="id006703894185413486">Yes
|
||
No
|
||
Yes
|
||
Yes
|
||
</pre></div><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id003030451618641431" id="id0012268541365229912" class="input-output-copier">Copy</div></div><pre id="id003030451618641431">10 8 6 11
|
||
1 2 4 7 8 9 11 13 19 20
|
||
2 13
|
||
5 8
|
||
8 1
|
||
6 15
|
||
1 15
|
||
2 7
|
||
7 6
|
||
8 9
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id00007163596982270759" id="id0004696282416006514" class="input-output-copier">Copy</div></div><pre id="id00007163596982270759">Yes
|
||
Yes
|
||
No
|
||
Yes
|
||
Yes
|
||
Yes
|
||
Yes
|
||
Yes
|
||
</pre></div><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id004021868626461702" id="id006245115827740666" class="input-output-copier">Copy</div></div><pre id="id004021868626461702">6 9 6 6
|
||
1 2 4 9 18 19
|
||
2 17
|
||
1 18
|
||
5 4
|
||
2 11
|
||
5 17
|
||
6 8
|
||
4 3
|
||
3 3
|
||
6 6
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id0028814164650272367" id="id003382640439163931" class="input-output-copier">Copy</div></div><pre id="id0028814164650272367">Yes
|
||
Yes
|
||
Yes
|
||
Yes
|
||
Yes
|
||
Yes
|
||
No
|
||
No
|
||
Yes
|
||
</pre></div><div class="input"><div class="title">Input<div title="Copy" data-clipboard-target="#id0036202103055617885" id="id007376883974196634" class="input-output-copier">Copy</div></div><pre id="id0036202103055617885">4 1 1 10
|
||
1 8 10 19
|
||
2 1
|
||
</pre></div><div class="output"><div class="title">Output<div title="Copy" data-clipboard-target="#id008954713496747658" id="id0012921296784619696" class="input-output-copier">Copy</div></div><pre id="id008954713496747658">Yes
|
||
</pre></div></div></div><div class="note"><div class="section-title">Note</div><p>Explanation of the first example:</p><p>In the first testcase the destination rock is the same as the starting rock, thus no jumps are required to reach it.</p><p>In the second testcase the frog can jump any distance in the range <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1495"><span class="MJXp-mo" id="MJXp-Span-1496" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-1497">5</span><span class="MJXp-mo" id="MJXp-Span-1498" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-1499">2</span><span class="MJXp-mo" id="MJXp-Span-1500" style="margin-left: 0em; margin-right: 0.222em;">;</span><span class="MJXp-mn" id="MJXp-Span-1501">5</span><span class="MJXp-mo" id="MJXp-Span-1502" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-1503">2</span><span class="MJXp-mo" id="MJXp-Span-1504" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-235">[5 - 2; 5 + 2]</script>. Thus, it can reach rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1505"><span class="MJXp-mn" id="MJXp-Span-1506">5</span></span></span><script type="math/tex" id="MathJax-Element-236">5</script> (by jumping <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1507"><span class="MJXp-mn" id="MJXp-Span-1508">7</span></span></span><script type="math/tex" id="MathJax-Element-237">7</script> to the right) and rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1509"><span class="MJXp-mn" id="MJXp-Span-1510">3</span></span></span><script type="math/tex" id="MathJax-Element-238">3</script> (by jumping <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1511"><span class="MJXp-mn" id="MJXp-Span-1512">3</span></span></span><script type="math/tex" id="MathJax-Element-239">3</script> to the left). From rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1513"><span class="MJXp-mn" id="MJXp-Span-1514">3</span></span></span><script type="math/tex" id="MathJax-Element-240">3</script> it can reach rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1515"><span class="MJXp-mn" id="MJXp-Span-1516">2</span></span></span><script type="math/tex" id="MathJax-Element-241">2</script> (by jumping <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1517"><span class="MJXp-mn" id="MJXp-Span-1518">5</span></span></span><script type="math/tex" id="MathJax-Element-242">5</script> to the left). From rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1519"><span class="MJXp-mn" id="MJXp-Span-1520">2</span></span></span><script type="math/tex" id="MathJax-Element-243">2</script> it can reach rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1521"><span class="MJXp-mn" id="MJXp-Span-1522">1</span></span></span><script type="math/tex" id="MathJax-Element-244">1</script> (by jumping <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1523"><span class="MJXp-mn" id="MJXp-Span-1524">4</span></span></span><script type="math/tex" id="MathJax-Element-245">4</script> to the left). However, there is no way to reach rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1525"><span class="MJXp-mn" id="MJXp-Span-1526">7</span></span></span><script type="math/tex" id="MathJax-Element-246">7</script>.</p><p>In the third testcase the frog can jump any distance in the range <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1527"><span class="MJXp-mo" id="MJXp-Span-1528" style="margin-left: 0em; margin-right: 0em;">[</span><span class="MJXp-mn" id="MJXp-Span-1529">5</span><span class="MJXp-mo" id="MJXp-Span-1530" style="margin-left: 0.267em; margin-right: 0.267em;">−</span><span class="MJXp-mn" id="MJXp-Span-1531">3</span><span class="MJXp-mo" id="MJXp-Span-1532" style="margin-left: 0em; margin-right: 0.222em;">;</span><span class="MJXp-mn" id="MJXp-Span-1533">5</span><span class="MJXp-mo" id="MJXp-Span-1534" style="margin-left: 0.267em; margin-right: 0.267em;">+</span><span class="MJXp-mn" id="MJXp-Span-1535">3</span><span class="MJXp-mo" id="MJXp-Span-1536" style="margin-left: 0em; margin-right: 0em;">]</span></span></span><script type="math/tex" id="MathJax-Element-247">[5 - 3; 5 + 3]</script>. Thus, it can reach rock number <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1537"><span class="MJXp-mn" id="MJXp-Span-1538">7</span></span></span><script type="math/tex" id="MathJax-Element-248">7</script> by jumping to rock <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1539"><span class="MJXp-mn" id="MJXp-Span-1540">5</span></span></span><script type="math/tex" id="MathJax-Element-249">5</script> first and to <span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1541"><span class="MJXp-mn" id="MJXp-Span-1542">7</span></span></span><script type="math/tex" id="MathJax-Element-250">7</script> afterwards.</p><p>The fourth testcase is shown in the explanation for the second testcase.</p></div></div><p> </p></div>
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