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 = Array.new(n) do
  si = gets.chomp.chars.map! { |c| c == '(' ? 1 : -1 }
  x = y = 0
  si.each { |k| x += k; y = x if x < y }
  [x, y]
end

if s.sum(&:first) != 0
  puts :No
  exit 0
end

# maximize k
k = 0
loop do
  sp, s = s.partition { |x, y| x >= 0 && y >= -k }
  break if sp.empty?
  k += sp.sum(&:first)
end

sm, s = s.partition { |x, y| x == y }
k += sm.sum(&:first)

if s.any? { |x, y| y < -k }
  puts :No
  exit 0
end

s.sort_by! { |x, y| [y, x] }

until s.empty?
  x, y = s.pop
  if y < -k
    puts :No
    exit 0
  end
  k += x
end

puts :Yes

load __FILE__ unless $stdin.eof?