fix: resolve typecheck errors in cache, atcoder, cses, and usaco

Problem: lua typecheck flagged missing start_time field on ContestSummary;
ty flagged BeautifulSoup Tag/NavigableString union on csrf_input.get(),
a 3-tuple unpack where _extract_problem_info now returns 4 values in
cses.py, and an untyped list assignment in usaco.py.

Solution: add start_time? to ContestSummary LuaDoc, guard csrf_input
with hasattr check and type: ignore, unpack precision from
_extract_problem_info in cses.py callers, and use cast() in usaco.py.

lua/cp/cache.lua

@@ -15,6 +15,7 @@
 ---@field display_name string
 ---@field name string
 ---@field id string
+---@field start_time? integer
 
 ---@class CombinedTest
 ---@field input string