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<iostream>
using namespace std;
#include<string>
#include<vector>
#include<map>


int main(){
	ios::sync_with_stdio(false);
	cin.tie(0);
	int n;
	cin >> n;
	int sub = 0;
	vector<vector<int>>v;
	map<int, vector<int>>m;

	for (int i = 0; i < n; i++){
		string s;
		cin >> s;
		sub = 0;
		//cout << s.length();
		v.push_back({});
		for (int j = 0; j < s.length(); j++){
			if (s[j] == '('){
				sub--; v[i].push_back(-1); }
			else 
			{ sub++; v[i].push_back(1); }
		}m[sub].push_back(i);
	}

	//map<int, int>m;
	map<int,vector<int>>::iterator iter;
	//m[3] = 1;
	//m[5] = 2;
	int b = 0;
	bool ans = 1;
	for (iter = m.begin(); iter != m.end(); iter++){
		//cout << iter->first;
		for (int k = 0; k < iter->second.size(); k++){
		for (int j = 0; j < v[(iter->second)[k]].size(); j++)
		{
			b = b + v[iter->second[k]][j];
			//cout << "\n" << b;
			if (b>0){ ans = 0; break; }
		}
	}
	}
	if (ans == 1 && b==0){
		cout << "Yes";
	}
	else cout << "No";
	//for (int)
	return 0;

}