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>
#include <vector>
#include <list>
#include <map>
#include <set>
#include <unordered_map>
#include <unordered_set>
#include <deque>
#include <stack>
#include <bitset>
#include <algorithm>
#include <functional>
#include <numeric>
#include <utility>
#include <sstream>
#include <iostream>
#include <fstream>
#include <string.h>
#include <iomanip>
#include <cstdio>
#include <cmath>
#include <cstdlib>
#include <ctime>
#include <queue>
#include <math.h>
#include <assert.h>
#define ff first
#define ss second
#define ll long long
#define mod 998244353
using namespace std;
ll n, gsum, nsum;
string s;
vector<pair<ll, ll> > vs, ns;
int main () {
    ios_base::sync_with_stdio(0);cin.tie(0);cout.tie(0);
    cin>>n;
    for(int i=1;i<=n;i++) {
        cin>>s;
        ll mi = 0, sum = 0;
        for(auto el : s) {
            if(el == ')')
                sum--;
            else
                sum++;
            mi = min(mi, sum);
        }
        if(mi < 0) {
            if(sum >= 0)
                vs.push_back({mi, sum});
            else {
                if(mi != sum) {
                    ns.push_back({mi, sum});
                }
                nsum += sum;
            }
        }
        else
            gsum += sum;
    }
    sort(vs.begin(), vs.end());
    reverse(vs.begin(), vs.end());
    sort(ns.begin(), ns.end());
    for(auto el : vs) {
        if(-gsum > el.first) {
            cout<<"No";
            return 0;
        }
        gsum += el.second;
    }
    for(auto el : ns) {
        if(-gsum > el.first) {
            cout<<"No";
            return 0;
        }
        gsum += el.second;
    }
    if(gsum == -nsum)
        cout<<"Yes";
    else
        cout<<"No";
}
/*
 2
 ((
 )))(
 */