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: Time Limit Exceeded)

# (()(()))  level
# (      ) --  0
#  ()(  )  -- -1
#     ()   -- -2
# 変化を計算して、
# - 最後0で終わる
# - 途中で0より大きくなることがない
# 変化量の合計は並べる順番に依存しない。
# 0より大きくならないようにするには、できるだけ深く潜るようにすればいい。

N = gets.to_i
LevelInfo = N.times.map do
  s = gets.chomp
  level = 0
  highest = 0
  s.each_char do |c|
    if c == '('
      level -= 1
    else
      level += 1
      highest = level if level > highest
    end
  end
  [level, highest]
end

# 変化の合計量のチェック
level_sum = LevelInfo.map{|level, highest| level}.sum()
if level_sum != 0
  puts 'No'
  exit(0)
end

# できるだけ0より大きくならないように並べ替える
downs = LevelInfo.select{|level, highest| level < 0}.sort_by{|level, highest| highest}
equals = LevelInfo.select{|level, highest| level == 0}
ups = LevelInfo.select{|level, highest| level > 0}.sort_by{|level, highest| - highest + level}

level_sum = 0
[downs, equals, ups].each do |sub|
  sub.each do |level, highest|
    if level_sum + highest > 0
      puts 'No'
      exit(0)
    end
    level_sum += level
  end
end

puts 'Yes'