#include <bits/stdc++.h>
using namespace std;
long long bs(vector<long long>& v, long long x) {
  long long l = 0;
  long long r = v.size();
  long long ans = r;
  while (l < r) {
    long long mid = (l + r) >> 1;
    if (x + v[mid] < 0) {
      ans = mid;
      r = mid;
    } else {
      l = mid + 1;
    }
  }
  return ans;
}
signed main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  long long n;
  cin >> n;
  vector<string> s(n);
  vector<map<long long, vector<long long>>> mp(n);
  vector<vector<long long>> v(n);
  vector<long long> sm(n);
  for (long long i = 0; i < n; i++) cin >> s[i];
  for (long long i = 0; i < n; i++) {
    long long smm = 0;
    long long mi = 0;
    for (long long j = 0; j < s[i].length(); j++) {
      char c = s[i][j];
      if (c == '(')
        smm++;
      else
        smm--;
      mp[i][smm].push_back(j);
      mi = min(mi, smm);
      v[i].push_back(mi);
    }
    sm[i] = smm;
  }
  vector<long long> SM(1 << n);
  for (long long i = 0; i < 1 << n; i++) {
    for (long long j = 0; j < n; j++) {
      if ((i >> j) & 1) {
        SM[i] += sm[j];
      }
    }
  }
  vector<vector<long long>> dp(1 << n, vector<long long>(2, INT_MIN));
  dp[0][0] = 0;
  dp[0][1] = 0;
  for (long long i = 1; i < 1 << n; i++) {
    for (long long j = 0; j < n; j++) {
      if ((i >> j) & 1) {
        long long mask = i ^ (1 << j);
        if (v[j].back() + SM[mask] >= 0) {
          auto it = mp[j].find(-SM[mask]);
          long long tot = 0;
          if (it != mp[j].end()) tot = it->second.size();
          dp[i][0] = max(dp[i][0], dp[mask][0] + tot);
        } else {
          long long idx = bs(v[j], SM[mask]);
          long long tot = 0;
          auto it = mp[j].find(-SM[mask]);
          if (it != mp[j].end()) {
            tot = lower_bound(it->second.begin(), it->second.end(), idx) -
                  it->second.begin();
          }
          dp[i][1] = max(dp[i][1], dp[mask][0] + tot);
        }
        dp[i][1] = max(dp[i][1], dp[mask][1]);
      }
    }
  }
  cout << max(dp[(1 << n) - 1][0], dp[(1 << n) - 1][1]);
  return 0;
}

### Prompt

Develop a solution in Cpp to the problem described below:

A bracket sequence is a string containing only characters "(" and ")". A regular bracket sequence (or, shortly, an RBS) 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 "()()" and "(())" are regular (the resulting expressions are: "(1)+(1)" and "((1+1)+1)"); 
  * bracket sequences ")(", "(" and ")" are not. 



Let's denote the concatenation of two strings x and y as x+y. For example, "()()" + ")(" = "()())(".

You are given n bracket sequences s_1, s_2, ..., s_n. You can rearrange them in any order (you can rearrange only the strings themselves, but not the characters in them).

Your task is to rearrange the strings in such a way that the string s_1 + s_2 + ... + s_n has as many non-empty prefixes that are RBS as possible.

Input

The first line contains a single integer n (1 ≤ n ≤ 20).

Then n lines follow, the i-th of them contains s_i — a bracket sequence (a string consisting of characters "(" and/or ")". All sequences s_i are non-empty, their total length does not exceed 4 ⋅ 10^5.

Output

Print one integer — the maximum number of non-empty prefixes that are RBS for the string s_1 + s_2 + ... + s_n, if the strings s_1, s_2, ..., s_n can be rearranged arbitrarily.

Examples

Input


2
(
)


Output


1


Input


4
()()())
(
(
)


Output


4


Input


1
(())


Output


1


Input


1
)(()


Output


0

Note

In the first example, you can concatenate the strings as follows: "(" + ")" = "()", the resulting string will have one prefix, that is an RBS: "()".

In the second example, you can concatenate the strings as follows: "(" + ")" + "()()())" + "(" = "()()()())(", the resulting string will have four prefixes that are RBS: "()", "()()", "()()()", "()()()()".

The third and the fourth examples contain only one string each, so the order is fixed.

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
long long bs(vector<long long>& v, long long x) {
  long long l = 0;
  long long r = v.size();
  long long ans = r;
  while (l < r) {
    long long mid = (l + r) >> 1;
    if (x + v[mid] < 0) {
      ans = mid;
      r = mid;
    } else {
      l = mid + 1;
    }
  }
  return ans;
}
signed main() {
  ios_base::sync_with_stdio(0);
  cin.tie(0);
  cout.tie(0);
  long long n;
  cin >> n;
  vector<string> s(n);
  vector<map<long long, vector<long long>>> mp(n);
  vector<vector<long long>> v(n);
  vector<long long> sm(n);
  for (long long i = 0; i < n; i++) cin >> s[i];
  for (long long i = 0; i < n; i++) {
    long long smm = 0;
    long long mi = 0;
    for (long long j = 0; j < s[i].length(); j++) {
      char c = s[i][j];
      if (c == '(')
        smm++;
      else
        smm--;
      mp[i][smm].push_back(j);
      mi = min(mi, smm);
      v[i].push_back(mi);
    }
    sm[i] = smm;
  }
  vector<long long> SM(1 << n);
  for (long long i = 0; i < 1 << n; i++) {
    for (long long j = 0; j < n; j++) {
      if ((i >> j) & 1) {
        SM[i] += sm[j];
      }
    }
  }
  vector<vector<long long>> dp(1 << n, vector<long long>(2, INT_MIN));
  dp[0][0] = 0;
  dp[0][1] = 0;
  for (long long i = 1; i < 1 << n; i++) {
    for (long long j = 0; j < n; j++) {
      if ((i >> j) & 1) {
        long long mask = i ^ (1 << j);
        if (v[j].back() + SM[mask] >= 0) {
          auto it = mp[j].find(-SM[mask]);
          long long tot = 0;
          if (it != mp[j].end()) tot = it->second.size();
          dp[i][0] = max(dp[i][0], dp[mask][0] + tot);
        } else {
          long long idx = bs(v[j], SM[mask]);
          long long tot = 0;
          auto it = mp[j].find(-SM[mask]);
          if (it != mp[j].end()) {
            tot = lower_bound(it->second.begin(), it->second.end(), idx) -
                  it->second.begin();
          }
          dp[i][1] = max(dp[i][1], dp[mask][0] + tot);
        }
        dp[i][1] = max(dp[i][1], dp[mask][1]);
      }
    }
  }
  cout << max(dp[(1 << n) - 1][0], dp[(1 << n) - 1][1]);
  return 0;
}

```