|Codeforces Round #604 (Div. 1)|
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:
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.
The only line contains a non-empty string consist of only '(', ')' and '?'. The length of the string is at most $$$2000$$$.
Print the answer modulo $$$998244353$$$ in a single line.
In the first test case, we can obtain $$$4$$$ bracket sequences by replacing all characters '?' with either '(' or ')':
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 ')':
So, the answer is $$$9 = 2 + 2 + 3 + 2$$$.