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 = N.times.map{ 
    s = gets.chomp
    {
        raw: s, 
        value: s.chars.inject({min: 0, last: 0}){|result, char|
            result[:last] += (char == '(' ? 1 : -1)
            result[:min] = [result[:min], result[:last]].min
            result
        }
    }
}
S.each do |s|
    s[:startable] = s[:raw][0] == "(" && s[:value][:min] >= 0 && s[:value][:last] >= 0
end
start = S.select{|s| s[:startable]}.max_by{|s| s[:value][:last]}
if start.nil?
    puts :No
    exit
end
current = start[:value][:last]
S.delete(start)
sorted = S.sort_by{|s| -s[:value][:last]}
while !sorted.empty?
    idx = sorted.find_index{|e| current + e[:value][:min] >= 0}
    if idx.nil?
        puts :No
        exit
    end
    s = sorted.delete_at(idx)
    current += s[:value][:last]
    if current < 0
        puts :No
        exit
    end
end
if current == 0
    puts :Yes
else
    puts :No
end