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: Accepted)

import sys
input=sys.stdin.readline

def count(s):
    m=0
    e=0
    for t in s:
        if t=='(':
            e+=1
        else:
            e-=1
            m=min(m,e)
    return (e,m)
def count_reverse(s):
    m=0
    e=0
    for t in s[::-1]:
        if t==')':
            e+=1
        else:
            e-=1
            m=min(m,e)
    return (e,m)

def main():
    n=int(input())
    S=[input().strip() for _ in range(n)]
    P,M=[],[]
    Z=[0]
    for s in S:
        e,m=count(s)
        if e>0:
            P.append((m,e))
        elif e<0:
            e,m=count_reverse(s)
            M.append((m,e))
        else:
            Z.append(m)
    P.sort(reverse=True); M.sort(reverse=True)
    s_p,s_m=0,0
    for m,e in P:
        if s_p+m<0:
            return 'No'
        s_p+=e
    for m,e in M:
        if s_m+m<0:
            return 'No'
        s_m+=e
    if s_p==s_m and s_p+min(Z)>=0:
        return 'Yes'
    return 'No'

if __name__=='__main__':
    print(main())