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: Accepted)

N = int(input())
S = [input() for i in range(N)]

xs1total = 0 # ((
xs2 = [] # )((
xs3 = [] # ))(
xs4total = 0 # ))

for s in S:
    n = 0
    m = 0
    for c in s:
        if c == "(":
            n += 1
        elif c == ")":
            n -= 1
            m = min(m, n)
    k = n - m
    if m == 0 and k == 0:
        continue
    elif m == 0:
        xs1total += k
    elif k == 0:
        xs4total += -m
    else:
        if -m < k:
            xs2.append((-m, k))
        else:
            xs3.append((-m, k))

xs2.sort()
xs3.sort(key=lambda x:-(x[1] - x[0]))

def check(xs1total, xs2, xs3, xs4total):
    t = xs1total
    for x in xs2:
        if t < x[0]:
            return False
        t = t - x[0] + x[1]
    for x in xs3:
        if t < x[0]:
            return False
        t = t - x[0] + x[1]
    return t == xs4total

#print(xs1total, xs2, xs3, xs4total)

if check(xs1total, xs2, xs3, xs4total):
    print("Yes")
else:
    print("No")