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: Wrong Answer)

N = int(input())
s = [input() for i in range(N)]
S = [[0]*len(s[i]) for i in range(N)]
for i, x in enumerate(s):
    res = []
    q = 0
    for j in range(len(x)):
        if x[j] == ')':
            S[i][j] = -1
        else:
            S[i][j] = 1
s = [0]*N
m = [len(S[i]) for i in range(N)]
for i, x in enumerate(S):
    tmp = 0
    for j in range(len(x)):
        tmp += S[i][j]
        m[i] = min(tmp, m[i])
    s[i] = tmp
if sum(s) == 0:
    ss = [(m[i], s[i]) for i in range(N)]
    ss.sort(key=lambda x: (x[0], x[1]), reverse=True)
    tmp = 0
    for x, y in ss:
        if tmp + x < 0:
            print("No")
            exit(0)
        tmp += y

    print("Yes")

else:
    print("No")