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: Accepted)

n = gets.chomp.to_i
akes = 0
tojis = 0
tojiake = []
n.times{|i|
  ake_count = 0
  toji_count = 0
  gets.chomp.split('').each{|v|
    if v == '('
      ake_count += 1
    else
      if ake_count > 0
        ake_count -= 1
      else
        toji_count += 1
      end
    end
  }

  if toji_count == 0
    akes += ake_count
  elsif ake_count == 0
    tojis += toji_count
  else
    tojiake.push([toji_count, ake_count])
  end
}

hiraki = akes

success = true
# p tojiake.sort{|a, b| (b[1] - b[0]) <=> (a[1] - a[0]) }
tojiake.sort{|a, b| (b[1] - b[0]) <=> (a[1] - a[0]) }.each{|i|
  hiraki -= i[0]
  if hiraki < 0
    success = false
    break
  end
  hiraki += i[1]
}
success = false if hiraki != tojis

puts success ? 'Yes' : 'No'