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
rr = ll = 0
ary = []

# p '('.ord #=> 40

n.times do
  s = gets.to_s.chomp.bytes
  # nil while s = s.sub("()", "")
  l = r = 0
  s.each do |c|
    if c == 40 # p '('.ord #=> 40
      l += 1
    elsif l == 0
      r += 1
    else
      l -= 1
    end
  end
  
  if l == 0
    rr += r
  elsif r == 0
    ll += l
  else
    ary << [r, l]
  end
end

ll -= rr

ary.sort_by!{|(r,l)| r - l }.each do |(r,l)|
  ll -= r
  break if ll < 0
  ll += l
end

puts ll == 0 ? "Yes" : "No"