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
start_depth = 0
end_depth = 0
center_blocks= []
N.times{
	s = gets.chomp
	d = 0
	min_d = 0
	max_d = 0
	s.each_char{|c|
		if c == "("
			d += 1
			if d > max_d
				max_d = d
			end
		else
			d -= 1
			if d < min_d
				min_d = d
			end
		end
	}
	depth = [-min_d, d - min_d]
	next if depth[0] == 0 && depth[1] == 0
	if depth[0] == 0
		start_depth += depth[1]
	elsif depth[1] == 0
		end_depth += depth[0]
	else
		center_blocks << depth
	end
}
loop{
	prev_size = center_blocks.size
	center_blocks.delete_if{|block|
		r = false
		if start_depth >= block[0]
			r = true
			start_depth += block[1] - block[0]
		elsif end_depth >= block[0]
			r = true
			end_depth += block[0] - block[1]
		end
		r
	}
	break if prev_size == center_blocks.size
}
if center_blocks.empty? && start_depth == end_depth
	puts "Yes"
else
	puts "No"
end