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;
 
#define ll          long long int
#define ld          long double
#define pb          push_back
#define mp          make_pair
#define ff          first
#define se          second
#define endl        '\n'
#define fd(n)       fixed<<setprecision(n)
#define MAX         100000000000000000
#define MAXN        300005
#define mod         998244353  
 
 
int main(){
    ios_base::sync_with_stdio(false);   
    cin.tie(NULL);
    ll n,i,count=0,sum;
    cin>>n;
    string s,s1="";
    vector<pair<ll,string> > vi;
    for(i=0;i<n;i++){
        cin>>s;
        count=0;
        for(auto j:s){
            if(j=='('){
                count++;
            }
            else{
                count--;
            }
        }
        sum+=count;
        vi.pb({count,s});
    }
    if(sum!=0){
        cout<<"No";
    }
    else{
        sort(vi.rbegin(),vi.rend());
        for(auto x:vi){
            s1+=x.se;
        }
        count=0;
        for(auto x:s1){
            if(x=='('){
                count++;
            }
            else{
                count--;
            }
            if(count<0){
                cout<<"No";
                return 0;
            }
        }
        cout<<"Yes";
    }
}