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
kakko_hiraku = 40
kakko_tojiru = 41 
l0_count = 0
r0_count = 0
others_left = 0
others_right = 0
n.times do |i|
  line = gets.chomp.codepoints
  left = 0
  right = 0
  line.each do |c|
    if c == kakko_hiraku
      right += 1
    else
      if right > 0
        right -= 1
      else
        left += 1
      end
    end
  end
  if left == 0 && right == 0
    next
  end
  if left == 0
    l0_count += right
  elsif right == 0
    r0_count += left
  else
    others_left += left
    others_right += right
  end
end

if others_left == 0 && others_right == 0
  if l0_count == r0_count
    puts "Yes"
  else
    puts "No"
  end
  exit
end

if l0_count == 0
  puts "No"
  exit
end

if r0_count == 0
  puts "No"
  exit
end

if l0_count < others_left 
  puts "No"
  exit
end

if r0_count < others_right
  puts "No"
  exit
end

if l0_count + others_right == r0_count + others_left
  puts "Yes"
else
  puts "No"
end