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_info1 = []
S_info2 = []

tot = 0

for _ in range(N):
    s = input()
    score = 0
    maxv = 0
    minv = 0
    for c in s:
        if c == "(":
            score += 1
        else:
            score -= 1
        maxv = max(maxv, score)
        minv = min(minv, score)
    tot += score
    if score > 0:
        S_info1.append([s, score, maxv, minv])
    else:
        S_info2.append([s, score, maxv, minv])

if tot != 0:
    print("No")
else:
    S_info1.sort(key = lambda x:(-x[1], -x[3]))
    #print(S_info1)
    tot = 0
    flag = True
    for i in range(len(S_info1)):
        s, score, maxv, minv = S_info1[i]
        if tot < -minv:
            flag = False
        tot += score
    if not flag:
        print("No")
    else:
        S_info = S_info1 + S_info2
        flag = True
        tot = 0
        Maxv = 0
        Minv = 0
        Maxscore = 0
        for i in range(N):
            s, score, maxv, minv = S_info[i]
            tot += score
            Maxv = max(Maxv, maxv)
            Minv = min(Minv, minv)
            if score > 0:
                Maxscore += score
        if tot == 0 and Maxscore >= -Minv:
            print("Yes")
        else:
            print("No")