F - Parenthesis Checking

Score : $500$ points

Problem Statement

Let us define a correct parenthesis sequence as a string that satisfies one of the following conditions.

• It is an empty string.
• It is a concatenation of (, $A$, ), in this order, for some correct parenthesis sequence $A$.
• It is a concatenation of $A$, $B$, in this order, for some correct parenthesis sequences $A$ and $B$.

We have a string $S$ of length $N$ consisting of ( and ).

Given $Q$ queries $\text{Query}_1,\text{Query}_2,\ldots,\text{Query}_Q$, process them in order. There are two kinds of queries, with the following formats and content.

• 1 l r: Swap the $l$-th and $r$-th characters of $S$.
• 2 l r: Determine whether the contiguous substring from the $l$-th through the $r$-th character is a correct parenthesis sequence.

Constraints

• $1 \leq N,Q \leq 2 \times 10^5$
• $S$ is a string of length $N$ consisting of ( and ).
• $1 \leq l < r \leq N$
• $N,Q,l,r$ are all integers.
• Each query is in the format 1 l r or 2 l r.
• There is at least one query in the format 2 l r.

Input

Input is given from Standard Input in the following format:

$N$ $Q$

$S$

$\text{Query}_1$

$\text{Query}_2$

$\vdots$

$\text{Query}_Q$

Output

For each query in the format 2 l r, print Yes if the contiguous substring is a correct parenthesis sequence, and No otherwise, followed by a newline.

5 3
(())(
2 1 4
2 1 2
2 4 5

Yes
No
No

In the first query, (()) is a correct parenthesis sequence.

In the second query, (( is not a correct parenthesis sequence.

In the third query, )( is not a correct parenthesis sequence.

5 3
(())(
2 1 4
1 1 4
2 1 4

Yes
No

In the first query, (()) is a correct parenthesis sequence.

In the second query, $S$ becomes )()((.

In the third query, )()( is not a correct parenthesis sequence.

8 8
(()(()))
2 2 7
2 2 8
1 2 5
2 3 4
1 3 4
1 3 5
1 1 4
1 6 8

Yes
No
No