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 ≤ n ≤ 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
#include <bits/stdc++.h>
using namespace std;
const int maxn = 1e6 + 10;
char arr[maxn];
int nlft[maxn], nrgt[maxn];
int llft[maxn], lrgt[maxn];
int main() {
  int n;
  int chge = 0;
  int lft = 0, rgt = 0;
  int rlft = 0, rrgt = 0;
  scanf("%d", &n);
  scanf("%s", arr);
  for (int i = 0; i < n; i++) {
    if (arr[i] == '(') lft++;
    if (arr[i] == ')') rgt++;
    nlft[i] = lft;
    nrgt[i] = rgt;
    if (rgt - lft >= 3) {
      lft = rgt = 0;
      break;
    }
  }
  for (int i = n - 1; i >= 0; i--) {
    if (arr[i] == '(') rlft++;
    if (arr[i] == ')') rrgt++;
    llft[i] = rlft;
    lrgt[i] = rrgt;
    if (rlft - rrgt >= 3) {
      lft = rgt = 0;
      break;
    }
  }
  if (lft == rgt + 2) {
    for (int i = n - 1; i >= 0; i--) {
      if (nlft[i] >= nrgt[i] + 2 && arr[i] == '(') {
        chge++;
      } else if (nlft[i] < nrgt[i] + 2 && chge > 0) {
        break;
      }
    }
  } else if (rgt == lft + 2) {
    for (int i = 0; i < n; i++) {
      if (lrgt[i] >= llft[i] + 2 && arr[i] == ')') {
        chge++;
      } else if (lrgt[i] < llft[i] + 2 && chge > 0) {
        break;
      }
    }
  }
  printf("%d", chge);
}
