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: Accepted)

#define _CRT_SECURE_NO_WARNINGS

#include <cstdio>
#include <cstdlib>
#include <cmath>
#include <climits>
#include <cfloat>
#include <cstring>
#include <map>
#include <utility>
#include <set>
#include <iostream>
#include <memory>
#include <string>
#include <vector>
#include <list>
#include <random>
#include <algorithm>
#include <functional>
#include <sstream>
#include <complex>
#include <stack>
#include <queue>
#include <unordered_set>
#include <unordered_map>
#include <array>
#include <cassert>
#include <bitset>
#include <cstdint>
using namespace std;
using LL = long long;
using PP = pair<pair<LL, LL>, int>;
vector<string>SS;

int main() {
	int N;
	cin >> N;
	vector<PP>vec;
	LL sum = 0;
	for (int i = 0; i < N; i++) {
		string S;
		cin >> S;
		LL a = 0, x = 0;
		for (char c : S) {
			if (c == '(')x++;
			else x--;
			a = min(a, x);
		}
		a = max(a, a - x);
		vec.push_back({ {a,x},i });
		sum += x;
		SS.push_back(S);
	}
	sort(vec.begin(), vec.end(), [](const PP& p1, const PP& p2) {
		if (p1.first.second > 0 && p2.first.second < 0) {
			return true;
		}
		if (p1.first.second < 0 && p2.first.second > 0) {
			return false;
		}
		if (p1.first.second > 0 && p2.first.second > 0) {
			return p1.first.first > p2.first.first;
		}
		if (p1.first.second < 0 && p2.first.second < 0) {
			return p1.first.first < p2.first.first;
		}
		if (p1.first.second > 0) {
			return true;
		}
		if (p1.first.second < 0) {
			return false;
		}
		if (p2.first.second > 0) {
			return false;
		}
		if (p2.first.second < 0) {
			return true;
		}
		return false;
		});
	LL now = 0;
	int cnt = 0;
	for (int i = 0; i < N; ++i) {
		for (char c : SS[vec[i].second]) {
			if (c == '(') {
				now++;
			}
			else {
				now--;
			}
			if (now < 0) {
				cout << "No" << endl;
				return 0;
			}
			if (now == 0)cnt++;
		}
	}
	if (now != 0) {
		cout << "No" << endl;
		return 0;
	}
	cout << "Yes" << endl;
	return 0;
}