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: Wrong Answer)

#include <bits/stdc++.h>
using namespace std;

pair<int, int> getMinAndSum(const string &str) {
    int mn = 0, sum = 0;
    for(auto &c : str) {
        sum += (c == '(' ? 1 : -1);
        mn = min(mn, sum);
    }
    return {mn, sum};
}

void fail() {
    cout << "No" << '\n';
    exit(0);
}

int main() {
    int N;
    cin >> N;
    vector<pair<int, int>> pos, neg;
    for(int i = 0; i < N; ++i) {
        string S;
        cin >> S;
        auto tmp = getMinAndSum(S);
        if(tmp.second > 0) pos.push_back(tmp);
        else neg.push_back(tmp);
    }
    sort(pos.begin(), pos.end());
    sort(neg.begin(), neg.end(), [](auto a, auto b) {
        return a.second - a.first > b.second - b.first;
    });
    int now = 0;
    for(auto[mn, sum] : pos) {
        if(now + mn < 0) fail();
        now += sum;
    }
    for(auto[mn, sum] : neg) {
        if(now + mn < 0) fail();
        now += sum;
    }
    cout << (now == 0 ? "Yes" : "No") << '\n';
    return 0;
}