E. Almost Regular Bracket Sequence
time limit per test
3 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

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.

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.

Examples
Input
6
(((())

Output
3

Input
6
()()()

Output
0

Input
1
)

Output
0

Input
8
)))(((((

Output
0