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 Ruby (2.7.1) (judge verdict: Wrong Answer)

n=gets.to_i
ss=$<.map &:chomp
tsum=0
tpre=0
xs=[]
ss.each{|s|
  t=m=0
  s.bytes{|c|
    if c==40
      t+=1
    else
      t-=1
      m=t if t<m
    end
  }
  tsum+=t
  if m==0
    tpre+=t
  elsif t>m
    xs.push([-m,t])
  end
}
if tsum!=0
  puts 'No'
  exit
end
xs.sort!
puts xs.all?{|(m,t)| tpre>=m and tpre+=t } ? 'Yes' : 'No'