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
l=[]
n.times do
  s=gets.chomp
  sl=0
  sr=0
  s.each_char do |ss|
    if ss==")"
      if sr>0
        sr-=1
      else
        sl+=1
      end
    else
      sr+=1
    end
  end
  l << [sl,sr]
end
c=0
l.sort!
i=0
while i<n
  if l[0][0]==0
    c+=l[0][1]
  else
    break
  end
  i+=1
end
l=l[i..n-1]
l.size.times do |i|
  l[i]=[l[i][0]-l[i][1],l[i][0],l[i][1]]
end
l.sort!
l.each do |xxx,sl,sr|
  c-=sl
  if c<0
    puts "No"
    exit
  end
  c+=sr
end
if c==0
  puts "Yes"
else
  puts "No"
end