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

lc = rc = 0
xs = []
N.times do
    l = r = 0
    gets.chomp.each_char do |c|
        if c == ')'
            if r == 0
                l += 1
            else
                r -= 1
            end
        else
            r += 1
        end
    end
    if l == 0 && r == 0
        # nop
    elsif l == 0
        rc += r
    elsif r == 0
        lc += l
    elsif l+r!=0
        xs << [l,r]
    end
end

if lc + rc == 0
    puts 'No'
    exit 0
end

# lc = 99
# rc = 3
# xs = [[4,99],[3,4]]
as,bs = xs.partition {|l,r|r-l>0}
as = as.sort_by{|l,r|[l,-r]}
bs = as.sort_by{|l,r|[l,-r]}
as.each do |l,r|
    rc -= l
    if rc < 0
        puts 'No'
        exit 0
    end
    rc += r
end
bs.each do |l,r|
    rc -= l
    if rc < 0
        puts 'No'
        exit 0
    end
    rc += r
end

# p [as,bs,lc,rc]

if lc == rc
    puts 'Yes'
else
    puts 'No'
end