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> //Andrei Alexandru a.k.a Sho10
#define ll long long int
#pragma GCC optimize("O3")
#pragma GCC optimize("Ofast")
#define all(a) (a).begin(), (a).end()
#define sz size
#define f first
#define s second
#define pb push_back
#define er erase
#define in insert
#define mp make_pair
#define pi pair
#define rc(s) return cout<<s,0
#define endl '\n'
#define mod 1000000007
#define PI 3.14159265359
#define CODE_START  ios_base::sync_with_stdio(false);cin.tie(0);cout.tie(0);
using namespace std;
ll n;
string s[1000005];
bool cmp(string a,string b){
ll val1=0,val2=0,s1=0,s2=0;
for(ll i=0;i<a.size();i++)
{
    if(a[i]==')'){
        s1--;
    }else s1++;
    val1=min(val1,s1);
}
for(ll i=0;i<b.size();i++)
{
    if(b[i]==')'){
        s2--;
    }else s2++;
    val2=min(val2,s2);
}
return val1>val2;
}
int32_t main(){
CODE_START;
cin>>n;
for(ll i=0;i<n;i++)
{
    cin>>s[i];
}
sort(s,s+n,cmp);
ll sum=0;
for(ll i=0;i<n;i++)
{
    for(ll j=0;j<s[i].size();j++)
    {
        if(s[i][j]==')'){
            sum--;
        }else sum++;
        if(sum<0){
            rc("No");
        }
    }
}
if(sum!=0){
    rc("No");
}
for(ll i=0;i<n;i++)
{
    reverse(s[i].begin(),s[i].end());
    for(ll j=0;j<s[i].size();j++)
    {
        if(s[i][j]=='('){
            s[i][j]=')';
           }else s[i][j]='(';
    }
}
sort(s,s+n,cmp);
sum=0;
for(ll i=0;i<n;i++)
{
    for(ll j=0;j<s[i].size();j++)
    {
        if(s[i][j]==')'){
            sum--;
        }else sum++;
        if(sum<0){
            rc("No");
        }
    }
}
cout<<"Yes"<<endl;
}