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 check(s):
    pos = 0
    min_path = 0
    for chr in s:
        if chr == '(':
            pos += 1
        else:
            pos -= 1
            if min_path > pos:
                min_path = pos

    return min_path, pos  # p = 左側に ) が残っている, q = 右側に ( が残っている


n = int(input())
up_list = []
down_list = []
to_zero_down = 0


for _ in range(n):
    s = input()
    min_path, to = check(s)
    if to == 0:
        if to_zero_down > min_path:
            to_zero_down = min_path

    elif to > 0:
        up = to
        up_list.append((min_path, up))

    else:
        down = -to
        down_list.append((min_path, down))


up_list.sort()
down_list.sort()

count = 0
for min_path, up in up_list:
    if count + min_path < 0:
        print('No')
        exit()
    count += up

if count + to_zero_down < 0:
    print('No')
    exit()

for min_path, down in down_list:
    if count + min_path < 0:
        print('No')
        exit()
    count -= down

if count == 0:
    print('Yes')
else:
    print('No')