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
brackets = []
n.times do
  tmp = gets.chomp
  left = 0
  right = 0
  tmp.each_char do |c|
    if c == "("
      right += 1
    else
      if right > 0
        right -= 1
      else
        left += 1
      end
    end
  end
  brackets << [right, left]
end
current_left = 0
current_right = 0
brackets.each do |left, right|
  deduction_1_right = [current_left, right].min
  deduction_1_left = [current_right, left].min
  next_right_1 = current_right + right - deduction_1_right - deduction_1_left
  next_left_1 = current_left + left - deduction_1_left - deduction_1_right
  next_left_2 = current_left + left - deduction_1_left - deduction_1_right
  next_right_2 = current_right + right - deduction_1_right - deduction_1_left
  if next_left_1 + next_right_1 < next_left_2 + next_right_2
    current_right = next_right_1
    current_left = next_left_1
  else
    current_right = next_right_2
    current_left = next_left_2
  end
end
puts current_left == 0 && current_right == 0 ? "Yes" : "No"