Problem:
Score : 600 points

Problem Statement

A bracket sequence is a string that is one of the following:

An empty string;

The concatenation of (, A, and ) in this order, for some bracket sequence A ;

The concatenation of A and B in this order, for some non-empty bracket sequences A and B /

Given are N strings S_i. Can a bracket sequence be formed by concatenating all the N strings in some order?

Constraints

1 \leq N \leq 10^6

The total length of the strings S_i is at most 10^6.

S_i is a non-empty string consisting of ( and ).

Input

Input is given from Standard Input in the following format:

N
S_1
:
S_N

Output

If a bracket sequence can be formed by concatenating all the N strings in some order, print Yes; otherwise, print No.

Sample Input 1

2
)
(()

Sample Output 1

Yes

Concatenating (() and ) in this order forms a bracket sequence.

Sample Input 2

2
)(
()

Sample Output 2

No

Sample Input 3

4
((()))
((((((
))))))
()()()

Sample Output 3

Yes

Sample Input 4

3
(((
)
)

Sample Output 4

No

Solution in C++ (GCC 9.2.1) (judge verdict: Accepted)

#include "bits/stdc++.h"
using namespace std;
using ll = long long;
#define rep(i, begin, n) for (int i = begin; i < n; i++)
#define repe(i, begin, n) for (int i = begin; i <= n; i++)
#define repr(i, begin, n) for (int i = begin; i > begin - n; i--)
#define repre(i, begin, end) for (int i = begin; i >= end; i--)

template <class T>
inline bool chmax(T& a, T b) {
  if (a < b) {
    a = b;
    return 1;
  }
  return 0;
}

template <class T>
inline bool chmin(T& a, T b) {
  if (a > b) {
    a = b;
    return 1;
  }
  return 0;
}

const int inf = 1000000007;
const int MOD = 1000000007;
const long long INF = 1000000000000000007;

// -------------------------------------------------------
int N;

int main() {
  cin >> N;
  vector<pair<int, int>> pos, neg;
  rep(i, 0, N) {
    string s;
    cin >> s;

    int m = 0, sum = 0;
    int left = 0, right = 0;
    for (int j = 0; j < int(s.size()); ++j) {
      if (s[j] == '(') {
        ++left;
        ++sum;
      } else {
        ++m;
        --sum;
        if (left > right) {
          ++right;
        }
      }
    }
    m -= right;
    if (sum >= 0) {
      pos.push_back(make_pair(m, sum));
    } else {
      neg.push_back(make_pair(m + sum, sum));
    }
  }

  sort(pos.begin(), pos.end());
  sort(neg.begin(), neg.end(), greater<pair<int, int>>());

  int ss = 0;
  for (auto e : pos) {
    int mm = e.first;
    int ssum = e.second;
    if (mm > ss) {
      cout << "No" << endl;
      return 0;
    }
    ss += ssum;
  }

  for (auto e : neg) {
    int ssum = e.second;
    int mm = e.first - ssum;
    if (mm > ss) {
      cout << "No" << endl;
      return 0;
    }
    ss += ssum;
  }

  if (ss != 0) {
    cout << "No" << endl;
    return 0;
  }

  cout << "Yes" << endl;
}