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())

mem = [0] * N
for i in range(N):
    s = input()
    x = 0
    y = 0
    for j in range(len(s)):
        if s[j] == '(':
            x += 1
        else:
            x -= 1
            y = min(x, y)
    mem[i] = (x, y)

mem_a = sorted([ele for ele in mem if ele[0] > 0], key=lambda x: -x[1])
mem_b = sorted([ele for ele in mem if ele[0] <= 0], key=lambda x: -x[1])

ans = True
val = 0
for ele in mem_a:
    if val + ele[1] < 0:
        ans = False
    val += ele[0]
for ele in mem_b:
    if val + ele[1] < 0:
        ans = False
    val += ele[0]


YesNo = lambda tf: print('Yes') if tf else print('No')
YesNo(ans and val == 0)