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())
plus_data=[]
minus_data=[]
for i in range(N):
    s=list(input())
    dif=0
    maxi=0
    mini=0
    for j in range(len(s)):
        if s[j]=="(":
            dif+=1
        else:
            dif-=1
        maxi=max(maxi,dif)
        mini=min(mini,dif)
    if mini>=0:  
        plus_data.append((maxi,mini,dif))
    else:
        minus_data.append((maxi,mini,dif))
    
minus_data.sort(key=lambda x:(-x[1]))
flag=True
sums=0
for some in plus_data:
    sums+=some[2]
for some in minus_data:
    maxi,mini,dif=some
    if sums+mini<0:
        flag=False
        break
    else:
        sums+=dif
if flag:
    if sums==0:
        print("Yes")
    else:
        print("No")
else:
    print("No")