#P1095E. Almost Regular Bracket Sequence
Almost Regular Bracket Sequence
Description
You are given a bracket sequence $s$ consisting of $n$ opening '(' and closing ')' brackets.
A regular bracket sequence is a bracket sequence that can be transformed into a correct arithmetic expression by inserting characters '1' and '+' between the original characters of the sequence. For example, bracket sequences "()()", "(())" are regular (the resulting expressions are: "(1)+(1)", "((1+1)+1)"), and ")(" and "(" are not.
You can change the type of some bracket $s_i$. It means that if $s_i = $ ')' then you can change it to '(' and vice versa.
Your task is to calculate the number of positions $i$ such that if you change the type of the $i$-th bracket, then the resulting bracket sequence becomes regular.
The first line of the input contains one integer $n$ ($1 \le n \le 10^6$) — the length of the bracket sequence.
The second line of the input contains the string $s$ consisting of $n$ opening '(' and closing ')' brackets.
Print one integer — the number of positions $i$ such that if you change the type of the $i$-th bracket, then the resulting bracket sequence becomes regular.
Input
The first line of the input contains one integer $n$ ($1 \le n \le 10^6$) — the length of the bracket sequence.
The second line of the input contains the string $s$ consisting of $n$ opening '(' and closing ')' brackets.
Output
Print one integer — the number of positions $i$ such that if you change the type of the $i$-th bracket, then the resulting bracket sequence becomes regular.
Samples
6
(((())
3
6
()()()
0
1
)
0
8
)))(((((
0