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 PyPy3 (7.3.0) (judge verdict: Wrong Answer)

n = int(input())
L = []
R = []
for _ in range(n):
  s = input()
  btm = 0
  total = 0
  for c in s:
    if c == "(":
      total += 1
    else:
      total -= 1
    btm = min(btm, total)
  if total > 0:
    L.append((btm, total))
  else:
    R.append((btm, total))
    
L.sort(key=lambda x: (-x[0], -x[1]))
R.sort(key=lambda x: (-x[1], x[0]))
LR = L + R

crr_end = 0
for b, e in LR:
  if crr_end + b < 0:
    print("No")
    exit()
  crr_end += e

if crr_end == 0:
  print("Yes")
else:
  print("No")