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)

gets
re = /(?<p>\(\g<p>*\))/

ll = r = l = rr = 0
$<.each{|ln|
	s = ln.chomp.gsub(re,'')
	next if s.empty?

	if s[0] == ?(
		ll += s.size
	elsif s[-1] == ?)
		rr += s.size
	else
		i = s.index(?()
		# r), (l, i), (s.size-i
		r += i-l if l<i
		l = [l-i,0].max+s.size-i
	end
}

puts(ll<r||rr<l||ll+l!=rr+r ? 'No' : 'Yes')