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;
int n,ct[1010101],nw[1010101],sum,al;
string s;
vector<pair<int,int> >v;
int main()
{
	cin>>n;
	for(int i=0;i<n;i++)
	{
		cin>>s;
		int a=0;
		for(int j=0;j<s.size();j++)
		{
			if(s[j]=='(')
			{
				a++;
			}
			else
			{
				a--;
				if(a<=0)
				{
					nw[i]=max(nw[i],abs(a));
				}
			}
		}
		ct[i]=a;
		sum+=a;
	}
	if(sum!=0)
	{
		cout<<"No"<<endl;
		return 0;
	}
	for(int i=0;i<n;i++)
	{
		if(ct[i]>0)
		{
			al+=ct[i];
			if(nw[i]>0)
			{
				v.push_back(make_pair(nw[i],ct[i]));
				al-=ct[i];
			}
		}
	}
	sort(v.begin(),v.end());
	for(int i=0;i<v.size();i++)
	{
		if(al>=v[i].first)
		{
			al+=v[i].second;
		}
		else
		{
			cout<<"No"<<endl;
			return 0;
		}
	}
	for(int i=0;i<n;i++)
	{
		if(ct[i]==0)
		{
			if(al<nw[i])
			{
				cout<<"No"<<endl;
				return 0;
			}
		}
	}
	v.clear();
	for(int i=0;i<n;i++)
	{
		if(ct[i]<0)
		{
			if(nw[i]>0)
			{
				v.push_back(make_pair(nw[i],ct[i]));
			}
		}
	}
	sort(v.begin(),v.end());
	for(int i=0;i<v.size();i++)
	{
		if(al>=v[i].first)
		{
			al+=v[i].second;
		}
		else
		{
			cout<<"No"<<endl;
			return 0;
		}
	}
	cout<<"Yes"<<endl;
	return 0;
}