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())
    if n%2:
        print("No")
        return
    sl = [None]*n
    for i in range(n):
        height, mi = 0, 0
        for si in input():
            if si == "(":
                height += 1
                mi = min(mi, height)
            else:
                height -= 1
                mi = min(mi, height)
            
        sl[i] = [mi, height]
    slup = [si for si in sl if si[1] >= 0]
    slup.sort(key = lambda x: -x[0])
    sldown = [si for si in sl if si[1] < 0]
    sldown.sort(key = lambda x: x[0])
    h = 0
    for si in slup:
        if h < 0 or h+si[0] < 0:
            print("No")
            return
        h += si[1]
    for si in sldown:
        if h < 0 or h+si[0] < 0:
            print("No")
            return
        h += si[1]
    if h == 0:
        print("Yes")
    else:
        print("No")
        

if __name__ == "__main__":
    main()