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())
sss = [input() for _ in range(n)]

a = []
b = []
ss = 0
for i in sss:
    mi = 0
    ma = 0
    s = 0
    for j in i:
        if j == "(":
            s += 1
        else:
            s -= 1
        mi = min(mi, s)
        ma = max(ma, s)
    if s >= 0:
        a.append([mi, ma, s, i])
    else:
        b.append([mi, ma, s, i])
    ss += s

if ss != 0:
    print("No")
    exit()
a.sort(key=lambda x: x[0] * 100000000 - x[2])
b.sort(reverse=1, key=lambda x: x[2] * 100000000 - x[0])
s = 0
for _, _, _, i in a + b:
    for j in i:
        if j == "(":
            s += 1
        else:
            s -= 1
        if s < 0:
            print("No")
            exit()
print("Yes")