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)

N = int(input())
S = [input() for _ in range(N)]
# L = []
start = False
end = False
left_sum = 0
right_sum = 0
for i in S:
    l_join = 0
    r_join = 0
    for j in i:
        if j == '(':
            r_join += 1
        else:
            if r_join > 0:
                r_join -= 1
            else:
                l_join += 1
    if r_join == 0 and l_join == 0:
        continue
    left_sum += l_join
    right_sum += r_join
    # L.append((l_join, r_join))
    if r_join == 0:
        end = True
    if l_join == 0:
        start = True
if (left_sum == right_sum) and start and end:
    print('Yes')
else:
    print('No')