CF1015F 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.