#include <bits/stdc++.h>
using namespace std;
int main() {
  int n;
  string s;
  cin >> n >> s;
  string rs(s.rbegin(), s.rend());
  for (int i = 0; i < n; ++i) {
    if (rs[i] == '(') {
      rs[i] = ')';
    } else {
      rs[i] = '(';
    }
  }
  vector<int> pref(n + 1), suf(n + 1);
  vector<bool> okp(n + 1), oks(n + 1);
  okp[0] = oks[n] = true;
  for (int i = 0; i < n; ++i) {
    pref[i + 1] = pref[i] + (s[i] == '(' ? +1 : -1);
    okp[i + 1] = okp[i] & (pref[i + 1] >= 0);
    suf[n - i - 1] = suf[n - i] + (rs[i] == '(' ? +1 : -1);
    oks[n - i - 1] = oks[n - i] & (suf[n - i - 1] >= 0);
  }
  int ans = 0;
  for (int i = 0; i < n; ++i) {
    if (!okp[i] || !oks[i + 1]) {
      continue;
    }
    if (s[i] == '(') {
      if (pref[i] > 0 && pref[i] - 1 - suf[i + 1] == 0) {
        ++ans;
      }
    } else {
      if (pref[i] + 1 - suf[i + 1] == 0) {
        ++ans;
      }
    }
  }
  cout << ans << endl;
  return 0;
}

### Prompt

Create a solution in CPP for the following problem:

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

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
int main() {
  int n;
  string s;
  cin >> n >> s;
  string rs(s.rbegin(), s.rend());
  for (int i = 0; i < n; ++i) {
    if (rs[i] == '(') {
      rs[i] = ')';
    } else {
      rs[i] = '(';
    }
  }
  vector<int> pref(n + 1), suf(n + 1);
  vector<bool> okp(n + 1), oks(n + 1);
  okp[0] = oks[n] = true;
  for (int i = 0; i < n; ++i) {
    pref[i + 1] = pref[i] + (s[i] == '(' ? +1 : -1);
    okp[i + 1] = okp[i] & (pref[i + 1] >= 0);
    suf[n - i - 1] = suf[n - i] + (rs[i] == '(' ? +1 : -1);
    oks[n - i - 1] = oks[n - i] & (suf[n - i - 1] >= 0);
  }
  int ans = 0;
  for (int i = 0; i < n; ++i) {
    if (!okp[i] || !oks[i + 1]) {
      continue;
    }
    if (s[i] == '(') {
      if (pref[i] > 0 && pref[i] - 1 - suf[i + 1] == 0) {
        ++ans;
      }
    } else {
      if (pref[i] + 1 - suf[i + 1] == 0) {
        ++ans;
      }
    }
  }
  cout << ans << endl;
  return 0;
}

```