Bracket Sequences II

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 (()()).