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)

class ST(object):
    s: str
    leftdiff: int
    rightmax: int


def main():
    N = int(input())
    PM = []
    PP = []
    MP = []
    MM = []
    for i in range(N):
        s = input()
        st = ST()
        st.s = s

        left = 0
        right = 0
        rightmax = 0
        for c in s:
            if c == "(":
                left += 1
            else:
                right += 1
            rightmax = max(rightmax, right - left)
        st.leftdiff = (left - right)
        st.rightmax = rightmax

        if st.leftdiff > 0 and st.rightmax <= 0:
            PM.append(st)
        if st.leftdiff > 0 and st.rightmax > 0:
            PP.append(st)
        if st.leftdiff <= 0:
            MM.append(st)

        # if st.leftdiff <= 0 and st.rightmax <= 0:
        #     MM.append(st)
        # if st.leftdiff <= 0 and st.rightmax > 0:
        #     MP.append(st)

    nleft = 0
    for st in PM:
        nleft += st.leftdiff
    for st in sorted(PP, key=lambda _st: _st.rightmax):
        if nleft - st.rightmax < 0:
            print("No")
            return
        nleft += st.leftdiff

    MMS = list(sorted(MM, key=lambda _st: _st.leftdiff, reverse=True))
    used = [False] * len(MMS)
    for i in range(len(MMS)):
        for j in range(len(MMS)):
            if used[j]:
                continue
            if MMS[j].rightmax <= nleft:
                used[j] = True
                nleft += MMS[j].leftdiff
                break
        else:
            print("No")
            return
                
    if nleft == 0:
        print("Yes")
    else:
        print("No")


main()