As Famil Door’s birthday is coming, some of his friends (like Gabi) decided to buy a present for him. His friends are going to buy a string consisted of round brackets since Famil Door loves string of brackets of length n more than any other strings!
The sequence of round brackets is called valid if and only if:
Gabi bought a string s of length m (m≤n) and want to complete it to obtain a valid sequence of brackets of length n. He is going to pick some strings p and q consisting of round brackets and merge them in a string p+s+q, that is add the string p at the beginning of the string s and string q at the end of the string s.
Now he wonders, how many pairs of strings p and q exists, such that the string p+s+q is a valid sequence of round brackets. As this number may be pretty large, he wants to calculate it modulo 109+7.
First line contains n and m (1≤m≤n≤100000,n-m≤2000)− the desired length of the string and the length of the string bought by Gabi, respectively.
The second line contains string s of length m consisting of characters '(' and ')' only.
Print the number of pairs of string p and q such that p+s+q is a valid sequence of round brackets modulo 109+7.
4 1
(
4
4 4
(())
1
4 3
(((
0
In the first sample there are four different valid pairs:
In the second sample the only way to obtain a desired string is choose empty p and q.
In the third sample there is no way to get a valid sequence of brackets.