Bracket Sequences II

• Time limit: 1.00 s
• Memory limit: 512 MB

Your task is to calculate the number of valid bracket sequences of length $n$ when a prefix of the sequence is given.

Input

The first input line has an integer $n$.

The second line has a string of $k$ characters: the prefix of the sequence.

Output

Print the number of sequences modulo $10^9+7$.

Constraints

• $1 \le k \le n \le 10^6$
  

Example

Input:
6
(()

Output:
2

Explanation: There are two possible sequences: (())() and (()()).