1140 : Parenthesis

Description

Bobo has a balanced parenthesis sequence P=p1 p2…pn of length n and q questions.

The i-th question is whether P remains balanced after p_ai and p_bi swapped. Note that questions are individual so that they have no affect on others.

Parenthesis sequence S is balanced if and only if:

1. S is empty;
2. or there exists balanced parenthesis sequence A,B such that S=AB;
3. or there exists balanced parenthesis sequence S’ such that S=(S’).

Input

The input contains at most 30 sets. For each set:

The first line contains two integers n,q (2 ≤ n ≤ 10⁵, 1 ≤ q ≤ 10⁵).

The second line contains n characters p1 p2…pn.

The i-th of the last q lines contains 2 integers ai,bi (1 ≤ ai, bi ≤ n, ai ≠ bi).

Output

For each question, output “Yes” if P remains balanced, or “No” otherwise.

Sample

4 2
(())
1 3
2 3
2 1
()
1 2

No
Yes
No

Hint

Source

Author

ftiasch