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
S = (0...N).map{
  stack = []
  gets.chomp.split("").each{ |c|
    c = c == ")" ? -1 : +1
    if c == -1
      if stack[-1] == 1
        stack.pop
        next
      end
    end
    stack.push(c)
  }
  d = stack.reduce(0){ |a,b| a + (b < 0 ? b : 0) }
  u = stack.reduce(0){ |a,b| a + (b > 0 ? b : 0) }
  [d, u + d]
}

if S.map{ |a,b| b }.reduce(0,:+) != 0
  puts "No"; exit
end

S.sort_by!{ |a,b| -a }

U = S.select{ |a,b| b >= 0 }
D = S.select{ |a,b| b < 0 }.map{ |a,b| [-a,-b] }
D.reverse!
#p U, D

ua,da = [U,D].map{ |ss|
  x = 0
  ss.each{ |a,b|
    if x + a < 0
      puts "No"; exit
    end
    x += b
  }
  x
}
puts ua == da ? "Yes" : "No"