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 Python (3.8.2) (judge verdict: Time Limit Exceeded)

from functools import cmp_to_key

n = int(input())
pref = []
suff = []

for i in range(n):
	s = input()
	pos = neg = 0
	for ch in s:
		if ch == '(':
			pos += 1
		elif pos == 0:
			neg += 1
		else:
			pos -= 1
	if pos >= neg:
		pref.append([pos, neg])
	else:
		suff.append([pos, neg])

# def comp1(x, y):
# 	return x[1] - y[1]
#
#
# def comp2(x, y):
# 	return -x[0] + y[0]


# pref.sort( key=cmp_to_key(comp1))
# suff.sort( key=cmp_to_key(comp2))

pref.sort(key=lambda x: x[1])
suff.sort(key=lambda x: -x[0])


pref = pref + suff

totPos = 0

for pos, neg in pref:
	if neg > totPos:
		print("No")
		exit(0)
	totPos -= neg
	totPos += pos

if totPos != 0:
	print("No")
else:
	print("Yes")