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)

# bracket sequencing

import sys


def input(): return sys.stdin.readline().rstrip()


N = int(input())
data = []
for _ in range(N):
    plus = 0
    minus = 0
    sums = 0
    S = input()
    for i in range(len(S)):
        if S[i] == "(":
            sums += 1
            plus = max(plus, sums)
        else:
            sums -= 1
            minus = min(minus, sums)
    data.append((plus, minus, sums))
# dataの内部構造 = max+ ,min-, last_dif
res = 0
for x in data:
    res += x[-1]
if res != 0:
    print("No")
    sys.exit()
# res=0の条件を満たす下で考える
plus_que = []
minus_que = []
for x in data:
    if x[-1] >= 0:
        plus_que.append(x)
    else:
        minus_que.append(x)

plus_que.sort(key=lambda x: (-x[1]))
minus_que.sort(key=lambda x: (-x[0]-x[2]))
flag = True
now_value = 0
for x in plus_que:
    if now_value+x[1] < 0:
        flag = False
        break
    else:
        now_value += x[-1]

for y in plus_que:
    if now_value+y[1] < 0:
        flag = False
        break
    else:
        now_value += y[-1]
if flag:
    print("Yes")
else:
    print("No")