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
a = b = 0
ary = []

n.times do
    s = gets.chomp
    x = y = 0

    i = 0
    s.each_char do |c|
        c == ?) ? i += 1 : i -= 1
        x = i if x < i
    end

    i = 0
    s.each_char.reverse_each do |c|
        c == ?( ? i += 1 : i -= 1
        y = i if y < i
    end

    if y == 0
        a += x
    elsif x == 0
        b += y
    else
        ary << [x, y]
    end
end

ary.sort_by!{|e| e[0] - e[1]}

ary.each do |e|
    a -= e[0]
    break if a < 0
    a += e[1]
end

puts a == b ? "Yes" : "No"