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)

def main():
    import sys
    from collections import deque
    sys.setrecursionlimit(10**6)
    input = sys.stdin.readline
    N = int(input())
    p, m = 0, 0
    p_, m_ = 0, 0
    pq = deque([])
    mq = deque([])
    for n in range(N):
        s = input().strip()
        mini = 0
        dst = 0
        for ss in [1 if ss == '(' else -1 for ss in s]:
            dst += ss
            mini = min(dst, mini)
        
        if dst >= 0:
            pq.append((mini, dst))
        elif dst < 0:
            mq.append((mini, dst))
    pq = sorted(pq, reverse=True)
    mq = sorted(mq)
    p = 0
    for mini, dst in pq + mq:
        if - mini > p:
            print('No')
            exit()
        p += dst
    if p != 0:
        print('No')
    else:
        print('Yes')


if __name__ == '__main__':
    main()