CF1264D1.Beautiful Bracket Sequence (easy version)

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题目描述

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.

输入格式

The only line contains a non-empty string consist of only '(', ')' and '?'. The length of the string is at most $2000$ .

输出格式

Print the answer modulo $998244353$ in a single line.

输入输出样例

输入#1
```
??
```
输出#1
```
1
```
输入#2
```
(?(?))
```
输出#2
```
9
```

说明/提示

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 $9 = 2 + 2 + 3 + 2$ .

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