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())
S = [input() for _ in range(N)]

D = [[0] * 2 for _ in range(N)]

for i in range(N):
  s = S[i]
  ma = 0
  l = 0
  r = 0
  for j in s:
    if j == ")":
      r += 1
    else:
      l += 1
    ma = max(ma, r - l)
  D[i] = [l - r, ma, s]

cnt = 0
P = []
M = []
for x, y ,s in D:
  if x >= 0:
    if y <= 0:
      cnt += x
    else:
      P.append([x, y])
  else:
    M.append([x, y, s])

P.sort(key = lambda x:x[1])

for x, y in P:
  if y > cnt:
    print("No")
    exit()
  cnt += x

if len(M) == 0:
  if cnt == 0:
    print("Yes")
  else:
    print("No")
  exit()

M.sort(key = lambda x:x[1], reverse = True)

for i in range(len(M) -1, -1, -1):
  x, y, s= M[i]
  if x == -y:
    g = x
    M[i] = [0, 0, ""]
    break
else:
  print("No")
  exit()

for x, y, s in M:
  if cnt < y:
    print("No")
    exit()
  cnt += x

if cnt + g == 0:
  print("Yes")
else:
  print("No")