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 Lua (LuaJIT 2.1.0) (judge verdict: Wrong Answer)

N=io.read"*n"
io.read()
S={}
for i=1,N do table.insert(S,io.read())end

local ls=string.byte"("
local rs=string.byte")"

local lr={}

for i,s in ipairs(S)do
	local n
	repeat
		s,n=s:gsub("%(%)","")
	until(n==0)
	local l,r=s:match("%)*()%(*()")
	r=r-l
	l=l-1
	--[[
	local b={s:byte(1,#s)}
	local l,r=0,0
	for j,v in ipairs(b) do
		if(v==ls)then
			r=r+1
		else
			if(r>0)then r=r-1
			else l=l+1 end
		end
	end
	]]
	if(l~=0 or r~=0)then table.insert(lr,{l,r})end
end

if(#lr==0)then print"Yes"return end

do
local s=0
for _,v in ipairs(lr)do
	s=s+v[1]-v[2]
end
if(s~=0)then print"No"return end
end

local bl,br=-1,-1
for i,v in ipairs(lr)do
	if(v[1]==0)then bl=i end
	if(v[2]==0)then br=i end
end
if(bl==-1 or br==-1)then print"No"return end

local s=lr[bl][2]
table.remove(lr,math.max(bl,br))
table.remove(lr,math.min(bl,br))
table.sort(lr,function(a,b)return a[1]-a[2]<b[1]-b[2]end)

for i,v in ipairs(lr)do
	s=s-v[2]
	if(s<0)then print"No"return end
	s=s+v[1]
end

print"Yes"