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 <iostream>
#include <string>
#include <cstring>
#include <cmath>
#include <vector>
#include <queue>
#include <set>
#include <algorithm>
#include <map>
#include <fstream>
 
using namespace std;
 
typedef long long ll;
typedef vector<int> vi;
typedef vector<ll> vl;
typedef pair<int, int> ii;
typedef vector<ii> vii;
typedef priority_queue<int> pqi;
typedef priority_queue<ii> pqii;
typedef map<int, int> mii;
typedef map<int, string> mis;
typedef map<string, int> msi;
typedef set<int> si;
typedef set<ll> sl;
typedef map<ll, ll> mll;
typedef queue<int> qi;
typedef queue<ii> qii;
typedef vector<string> vs;
typedef pair<ll, ll> iil;
typedef vector<iil> viil;
 
#define pb push_back
#define mp make_pair
#define rep(i, n) for (int i = 0 ; i < n ; i++)
#define INF 1000000000
 
void solve() {
    int n;
    cin >> n;
    int sm = 0;
    vii b;
    rep(i, n) {
        string s;
        cin >> s;
        int ct = 0;
        int mb = 0;
        for (char c : s) {
            if (c == '(') {
                ct++;
            }
            else {
                ct--;
            }
            mb = min(mb, ct);
        }
        b.pb({ct, -mb});
        sm += ct;
    }
    if (sm != 0) {
        cout << "No" << endl;
        return;
    }
    sort(b.begin(), b.end());
    reverse(b.begin(), b.end());
    sm = 0;
    rep(i, n) {
        if (sm < b[i].second) {
            cout << "No" << endl;
            return;
        }
        sm += b[i].first;
    }
    cout << "Yes" << endl;
    
}
 
void querySolve() {
    int t;
    cin >> t;
    while (t--) {
        solve();
    }
}
 
int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(0);
    
    solve();
    //querySolve();
}