A11894. Bracket Substring

You are given a bracket sequence $s$ (not necessarily a regular one). A bracket sequence is a string containing only characters '(' and ')'.

A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters '1' and '+' between the original characters of the sequence. For example, bracket sequences "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)"), and ")(", "(" and ")" are not.

Your problem is to calculate the number of regular bracket sequences of length $2n$ containing the given bracket sequence $s$ as a substring (consecutive sequence of characters) modulo $10^9+7$ ( $1000000007$ ).

The first line of the input contains one integer $n$ ( $1 \le n \le 100$ ) — the half-length of the resulting regular bracket sequences (the resulting sequences must have length equal to $2n$ ).

The second line of the input contains one string $s$ ( $1 \le |s| \le 200$ ) — the string $s$ that should be a substring in each of the resulting regular bracket sequences ( $|s|$ is the length of $s$ ).

Print only one integer — the number of regular bracket sequences containing the given bracket sequence $s$ as a substring. Since this number can be huge, print it modulo $10^9+7$ ( $1000000007$ ).

All regular bracket sequences satisfying the conditions above for the first example:

- "(((()))())";
- "((()()))()";
- "((()))()()";
- "(()(()))()";
- "()((()))()".

All regular bracket sequences satisfying the conditions above for the second example:

- "((()))";
- "(()())";
- "(())()";
- "()(())".

And there is no regular bracket sequences of length $4$ containing "(((" as a substring in the third example.