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 sys
input = sys.stdin.readline

N = int(input())
S = [input().rstrip() for _ in range(N)]

def calc(S):
    now = 0
    ret = 0
    for s in S:
        if s == '(':
            now += 1
        else:
            now -= 1
        ret = min(ret, now)
    return ret

H = []
P = []
L = []

for s in S:
    less = -calc(s)
    if less == 0:
        H.append(s)
    else:
        l = s.count('(')
        r = s.count(')')
        if l > r:
            P.append((less, s))
        else:
            L.append((less, s))

P.sort(key=lambda a: a[0])
L.sort(key=lambda a: a[0], reverse=True)
P = [s for _, s in P]
L = [s for _, s in L]
ans = ''.join(H) + ''.join(P) + ''.join(L)

l = ans.count('(')
r = ans.count(')')

if l == r and calc(ans) == 0:
    print('Yes')
else:
    print('No')