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: Time Limit Exceeded)

import sys
input = sys.stdin.readline

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

A = []
B = []

for s in S:
    a = 0
    k = 0
    for i in range(len(s)):
        if s[i] == ')':
            k += 1
        else:
            k -= 1
        a = max(a, k)
    b = 0
    k = 0
    for i in range(len(s))[::-1]:
        if s[i] == '(':
            k += 1
        else:
            k -= 1
        b = max(b, k)
    A.append(a)
    B.append(b)

L = []
R = []

for i in range(N):
    if B[i] - A[i] >= 0:
        L.append((A[i], i))
    else:
        R.append((B[i], i))

L.sort()
R.sort(reverse=True)

T = ''

for a, i in L:
    T += S[i]

for b, i in R:
    T += S[i]

k = 0

for t in T:
    if t == '(':
        k += 1
    else:
        k -= 1
    if k < 0:
        print('No')
        break
else:
    if k == 0:
        print('Yes')
    else:
        print('No')