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: Runtime Error)

n,*s=open(0)
n=int(n)
f=g=h=a=b=0
for t in s:
    c=l=r=d=0
    for u in t[:-1]:
        if u=='(':
            if d<0:d=0
            d+=1
            r+=1
        else:
            d-=1
            if d>=0:c+=1
            l+=1
    x=r-c
    y=l-c
    if x==y==0:h+=1
    elif x==0:f=1
    elif y==0:g=1
    a+=x
    b+=y
if (f and g or h and (f or g) or h > 1) and a==b:
    print('Yes')
else:
    if a==b:1/0
    print('No')