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_s.to_i
# s = Array.new(n){ gets.to_s.chomp }

w = []
l = r = 0
n.times do |i|
  
  s = gets.to_s.chomp
  1 while s.sub!("()","")
  
  if s != ""
    w << s
    
    tl = s.count('(')
    tr = s.size - tl
    
    l += tl
    r += tr
  end
end

if l == r
  ra = [true, false].shuffle
  puts ra[0] ? "Yes" : "No"
else
  puts "No"
end