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 PyPy3 (7.3.0) (judge verdict: Wrong Answer)

def main():
    n = int(input())
    up, down = [], []
    for _ in range(n):
        s = list(input())
        m = 0
        height = 0
        for si in s:
            if si == "(":
                height += 1
            else:
                height -= 1
                m = min(m, height)
        if height >= 0:
            up.append([height, m])
        else:
            down.append([m-height, -height])
    up.sort(reverse=True)
    down.sort(reverse=True)
    if sum([i for i, j in up])-sum([j for i, j in down]) != 0:
        print("No")
        return
    h = 0
    for hi, mi in up:
        h += hi
    h = 0
    for hmi, hi in down:
        if h+hmi >= 0:
            h += hi
        else:
            print("No")
            return
    print("Yes")
    
        
if __name__ == "__main__":
    main()