CF1264D1 Beautiful Bracket Sequence (easy version)

This is the easy version of this problem. The only difference is the limit of $ n $ - the length of the input string. In this version, $ 1 \leq n \leq 2000 $ . The hard version of this challenge is not offered in the round for the second division. Let's define a correct bracket sequence and its depth as follow: - An empty string is a correct bracket sequence with depth $ 0 $ . - If "s" is a correct bracket sequence with depth $ d $ then "(s)" is a correct bracket sequence with depth $ d + 1 $ . - If "s" and "t" are both correct bracket sequences then their concatenation "st" is a correct bracket sequence with depth equal to the maximum depth of $ s $ and $ t $ . For a (not necessarily correct) bracket sequence $ s $ , we define its depth as the maximum depth of any correct bracket sequence induced by removing some characters from $ s $ (possibly zero). For example: the bracket sequence $ s = $ "())(())" has depth $ 2 $ , because by removing the third character we obtain a correct bracket sequence "()(())" with depth $ 2 $ . Given a string $ a $ consists of only characters '(', ')' and '?'. Consider all (not necessarily correct) bracket sequences obtained by replacing all characters '?' in $ a $ by either '(' or ')'. Calculate the sum of all the depths of all these bracket sequences. As this number can be large, find it modulo $ 998244353 $ . Hacks in this problem in the first division can be done only if easy and hard versions of this problem was solved.

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In the first test case, we can obtain $ 4 $ bracket sequences by replacing all characters '?' with either '(' or ')': - "((". Its depth is $ 0 $ ; - "))". Its depth is $ 0 $ ; - ")(". Its depth is $ 0 $ ; - "()". Its depth is $ 1 $ . So, the answer is $ 1 = 0 + 0 + 0 + 1 $ . In the second test case, we can obtain $ 4 $ bracket sequences by replacing all characters '?' with either '(' or ')': - "(((())". Its depth is $ 2 $ ; - "()()))". Its depth is $ 2 $ ; - "((()))". Its depth is $ 3 $ ; - "()(())". Its depth is $ 2 $ . So, the answer is $ 9 = 2 + 2 + 3 + 2 $ .