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 <stdio.h>
#include <algorithm>
#include <assert.h>
#include <cmath>
#include <deque>
#include <iostream>
#include <limits.h>
#include <map>
#include <math.h>
#include <queue>
#include <set>
#include <stdlib.h>
#include <string.h>
#include <string>
#include <unordered_map>
#include <unordered_set>
#include <vector>
#define ll long long
#define rep2(i,a,b) for(ll i=a;i<=b;++i)
#define rep(i,n) for(ll i=0;i<n;i++)
#define rep3(i,a,b) for(ll i=a;i>=b;i--)
#define REP(e,v) for(auto e:v)
#define pii pair<int,int>
#define pll pair<ll,ll>
#define mp make_pair
#define mt make_tuple
#define pq priority_queue<ll>
#define pqg priority_queue<int,vector<int>,greater<int>>
#define pb push_back
#define vec vector<int>
#define vecll vector<ll>
#define vecpii vector<pii>
#define vecpll vector<pll>
#define vecbl vector<bool>
#define endl "\n"
#define ALL(c) (c).begin(),(c).end()
using namespace std;
int in() {int x;scanf("%d",&x);return x;}
ll lin() {ll x;scanf("%lld",&x);return x;}
string stin(){string s;cin>>s;return s;}

bool func(){
    int n=in();
    vec r(n),l(n);
    int flagL=0,flagR=0;
    int tmp=0;
    for(int i=0;i<n;i++){
        string s=stin();
        rep3(j,s.length()-1,0){
            if(s[j]==')')l[i]++;
            else if(l[i]>0)l[i]--;
        }
        rep(j,s.length()){
            if(s[j]=='(')r[i]++;
            else if(r[i]>0)r[i]--;
        }
        tmp+=l[i];
        tmp-=r[i];
        if(l[i]==0 && r[i]==0)continue;
        if(flagL==0)flagL=-1;
        if(flagR==0)flagR=-1;
        if(l[i]==0)flagL=1;
        if(r[i]==0)flagR=1;
    }
    return tmp==0 && ((flagL==1 && flagR==1) || (flagL==0 && flagR==0));
}

int main(){
    string ans=func()?"Yes":"No";
    cout<<ans<<endl;
}