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, *Ss = readlines.map(&:chomp)

xs = Ss.map do |s|
  o, c = 0, 0
  s.chars.each do |x|
    if x == ')'
      if o == 0
        c += 1
      else
        o -= 1
      end
    else
      o += 1
    end
  end
  [c, o]
end.transpose

cs, os = 0, 0
xs.sort_by{|c, o| c - o}.each do |c, o|
  os -= c
  break if os < 0
  os += o
end
puts os == cs ? 'Yes' : 'No'