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 i in range(N)]

def brancket(s):
    ans = [0, 0]
    l, r = 0, 0
    for i in s:
        if i == ')':
            if r == 0:
                ans[0] += 1
            else:
                r -= 1
        if i == '(':
            r += 1
    ans[1] = r
    return ans

L, R, tl, tr = 0, 0, False, False
bigl, bigr = 0, 0
for i in S:
    if (a := brancket(i)) != [0, 0]:
        if a[0] == 0:
            tl = True
        if a[1] == 0:
            tr = True
    L += a[0]
    R += a[1]
    bigl = max(bigl, a[0])
    bigr = max(bigr, a[1])

if L != 0 or R != 0:
    if bigl <= R - bigr and bigr <= L - bigl:
        print('Yes' if tl and tr and L == R else 'No')
    else:
        print('No')
        
else:
    print('Yes')