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 Python (3.8.2) (judge verdict: Wrong Answer)

#import numpy as np
#import math
#from decimal import *
#from numba import njit

#@njit
def main():
    N = int(input())
    A = []
    for i in range(N):
        string = input()
        m = 0
        s = 0
        for c in string:
            if c == '(':
                s += 1
            else:
                s -= 1
                m = min(m, s)
        A += [m,s],
    
    A = sorted(A, key=lambda x: (x[0],x[1]), reverse=True)
    s = 0
    for i in range(len(A)):
        if A[i][0] < s or s +A[i][1] < 0:
            print('No')
            return
        s += A[i][1]
    
    if s == 0:
        print('Yes')
    else:
        print('No')

main()