You are an expert Python programmer. Write correct, efficient Python 3 code that solves the problem and passes all tests. Output the code in the following format: ```python
<code>
```

PROBLEM:
You are given a binary string s of length n and an integer numOps.
You are allowed to perform the following operation on s at most numOps times:

Select any index i (where 0 <= i < n) and flip s[i]. If s[i] == '1', change s[i] to '0' and vice versa.

You need to minimize the length of the longest substring of s such that all the characters in the substring are identical.
Return the minimum length after the operations.

Example 1:

Input: s = "000001", numOps = 1
Output: 2
Explanation:
By changing s[2] to '1', s becomes "001001". The longest substrings with identical characters are s[0..1] and s[3..4].

Example 2:

Input: s = "0000", numOps = 2
Output: 1
Explanation:
By changing s[0] and s[2] to '1', s becomes "1010".

Example 3:

Input: s = "0101", numOps = 0
Output: 1

Constraints:

1 <= n == s.length <= 105
s consists only of '0' and '1'.
0 <= numOps <= n

Your solution to the problem should be a method of the class Solution called minLength and should pass all tests. Use the following starter code:
```python
class Solution:
    def minLength(self, s: str, numOps: int) -> int:
        # Add your code here
```

SOLUTION:

Question:
PROBLEM:
You are given a binary string s of length n and an integer numOps.
You are allowed to perform the following operation on s at most numOps times:

Select any index i (where 0 <= i < n) and flip s[i]. If s[i] == '1', change s[i] to '0' and vice versa.

You need to minimize the length of the longest substring of s such that all the characters in the substring are identical.
Return the minimum length after the operations.

Example 1:

Input: s = "000001", numOps = 1
Output: 2
Explanation:
By changing s[2] to '1', s becomes "001001". The longest substrings with identical characters are s[0..1] and s[3..4].

Example 2:

Input: s = "0000", numOps = 2
Output: 1
Explanation:
By changing s[0] and s[2] to '1', s becomes "1010".

Example 3:

Input: s = "0101", numOps = 0
Output: 1

Constraints:

1 <= n == s.length <= 105
s consists only of '0' and '1'.
0 <= numOps <= n

Your solution to the problem should be a method of the class Solution called minLength and should pass all tests. Use the following starter code:
```python
class Solution:
    def minLength(self, s: str, numOps: int) -> int:
        # Add your code here
```

SOLUTION:

assert Solution().minLength(s = "000001", numOps = 1) == 2

assert Solution().minLength(s = "111000", numOps = 2) == 1

assert Solution().minLength(s = "11111111", numOps = 3) == 2

assert Solution().minLength(s = "111110", numOps = 2) == 1

assert Solution().minLength(s = "101010", numOps = 3) == 1

assert Solution().minLength(s = "1010101010", numOps = 5) == 1

assert Solution().minLength(s = "0101", numOps = 0) == 1

assert Solution().minLength(s = "0000", numOps = 2) == 1

assert Solution().minLength(s = "11111011", numOps = 3) == 1

assert Solution().minLength(s = "00110011", numOps = 3) == 2

assert Solution().minLength(s = "111111", numOps = 3) == 1

assert Solution().minLength(s = "00000000000000", numOps = 7) == 1

assert Solution().minLength(s = "1001001001001001", numOps = 5) == 2

assert Solution().minLength(s = "0000000000000000", numOps = 7) == 2

assert Solution().minLength(s = "0000111100001111", numOps = 8) == 1

assert Solution().minLength(s = "11111111111111", numOps = 8) == 1

assert Solution().minLength(s = "0000011111", numOps = 4) == 1

assert Solution().minLength(s = "1111000011110000", numOps = 4) == 2

assert Solution().minLength(s = "1111111111", numOps = 10) == 1

assert Solution().minLength(s = "1100110011", numOps = 4) == 2

assert Solution().minLength(s = "11001100110011", numOps = 6) == 2

assert Solution().minLength(s = "1100110011001100", numOps = 6) == 2

assert Solution().minLength(s = "001100110011", numOps = 5) == 2

assert Solution().minLength(s = "000110001100011", numOps = 5) == 2

assert Solution().minLength(s = "111000111000111", numOps = 6) == 1

assert Solution().minLength(s = "1111000000", numOps = 3) == 2

assert Solution().minLength(s = "110001110011", numOps = 4) == 2

Answer: assert Solution().minLength(s = "000001", numOps = 1) == 2

Answer: assert Solution().minLength(s = "111000", numOps = 2) == 1

Answer: assert Solution().minLength(s = "11111111", numOps = 3) == 2

Answer: assert Solution().minLength(s = "111110", numOps = 2) == 1

Answer: assert Solution().minLength(s = "101010", numOps = 3) == 1

Answer: assert Solution().minLength(s = "1010101010", numOps = 5) == 1

Answer: assert Solution().minLength(s = "0101", numOps = 0) == 1

Answer: assert Solution().minLength(s = "0000", numOps = 2) == 1

Answer: assert Solution().minLength(s = "11111011", numOps = 3) == 1

Answer: assert Solution().minLength(s = "00110011", numOps = 3) == 2

Answer: assert Solution().minLength(s = "111111", numOps = 3) == 1

Answer: assert Solution().minLength(s = "00010001000100010001", numOps = 3) == 3

Answer: assert Solution().minLength(s = "00000000001111111111", numOps = 3) == 5

Answer: assert Solution().minLength(s = "01010101010101010101010101010101", numOps = 0) == 1

Answer: assert Solution().minLength(s = "00000000000000000000000000000000000000000000000000000000000000000000000000000000", numOps = 100) == 1

Answer: assert Solution().minLength(s = "111111111100000000", numOps = 10) == 1

Answer: assert Solution().minLength(s = "00000000000000", numOps = 7) == 1

Answer: assert Solution().minLength(s = "111011101110111011101110111011101110111011101110", numOps = 20) == 1

Answer: assert Solution().minLength(s = "000100010001000100010001", numOps = 8) == 1

Answer: assert Solution().minLength(s = "000000000000000000000000000000000000000000000000", numOps = 50) == 1

Answer: assert Solution().minLength(s = "111100001111000011110000111100001111000011110000", numOps = 20) == 2

Answer: assert Solution().minLength(s = "1100110011001100110011001100110011001100110011001100", numOps = 15) == 2

Answer: assert Solution().minLength(s = "101010101010101010101010101010101010101010101010", numOps = 100) == 1

Answer: assert Solution().minLength(s = "111100001111000011110000", numOps = 12) == 1

Answer: assert Solution().minLength(s = "000000000000000000000000", numOps = 15) == 1

Answer: assert Solution().minLength(s = "111000111000111000111000111000111000111000111000", numOps = 20) == 1

Answer: assert Solution().minLength(s = "000111000111000111000111", numOps = 15) == 1

Answer: assert Solution().minLength(s = "000000000000000000000000", numOps = 20) == 1

Answer: assert Solution().minLength(s = "11110101010101010101", numOps = 5) == 1

Answer: assert Solution().minLength(s = "000111000111000111000111000111000111000111000111", numOps = 25) == 1

Answer: assert Solution().minLength(s = "0000000000111111111111111111111111111111111111111111", numOps = 25) == 2

Answer: assert Solution().minLength(s = "1001001001001001", numOps = 5) == 2

Answer: assert Solution().minLength(s = "11111111111111111111", numOps = 0) == 20

Answer: assert Solution().minLength(s = "0000000000000000", numOps = 7) == 2

Answer: assert Solution().minLength(s = "0000111100001111", numOps = 8) == 1

Answer: assert Solution().minLength(s = "00000000001111111111000000000011111111110000000000", numOps = 20) == 2

Answer: assert Solution().minLength(s = "000111000111000111000111000111000111000111000111", numOps = 20) == 1

Answer: assert Solution().minLength(s = "11111111111111", numOps = 8) == 1

Answer: assert Solution().minLength(s = "1111111111000000000011111111110000000000", numOps = 16) == 2

Answer: assert Solution().minLength(s = "0000011111", numOps = 4) == 1

Answer: assert Solution().minLength(s = "01010101010101010101010101", numOps = 12) == 1

Answer: assert Solution().minLength(s = "1111111111111111111111111111111111111111111111111111", numOps = 0) == 52

Answer: assert Solution().minLength(s = "11001100110011001100110011001100110011001100", numOps = 15) == 2

Answer: assert Solution().minLength(s = "11110000000011111111", numOps = 10) == 1

Answer: assert Solution().minLength(s = "00011100011100011100", numOps = 6) == 2

Answer: assert Solution().minLength(s = "01100110011001100110011001100110", numOps = 15) == 2

Answer: assert Solution().minLength(s = "0101010101010101010101010101010101010101010101010101010101010101", numOps = 20) == 1

Answer: assert Solution().minLength(s = "10000000000000000000", numOps = 10) == 1

Answer: assert Solution().minLength(s = "11110000111100001111", numOps = 8) == 2

Answer: assert Solution().minLength(s = "10101010101010101010101010101010101010101010101010101010101010101010101010101010", numOps = 150) == 1

Answer: assert Solution().minLength(s = "111100001111000011110000", numOps = 8) == 2

Answer: assert Solution().minLength(s = "10101010101010101010", numOps = 10) == 1

Answer: assert Solution().minLength(s = "111111111111111111111111111111111111111111111111", numOps = 30) == 1

Answer: assert Solution().minLength(s = "01010101010101010101010101010101010101010101010101", numOps = 25) == 1

Answer: assert Solution().minLength(s = "111000111000111000111", numOps = 5) == 3

Answer: assert Solution().minLength(s = "011001100110011001100110", numOps = 12) == 1

Answer: assert Solution().minLength(s = "101010101010101010101010", numOps = 10) == 1

Answer: assert Solution().minLength(s = "110011001100110011001100110011001100110011001100110011001100", numOps = 30) == 1

Answer: assert Solution().minLength(s = "000111000111000111", numOps = 4) == 3

Answer: assert Solution().minLength(s = "010101010101010101010101", numOps = 12) == 1

Answer: assert Solution().minLength(s = "00000000000000000000", numOps = 5) == 3

Answer: assert Solution().minLength(s = "111000111000111000111", numOps = 9) == 1

Answer: assert Solution().minLength(s = "000110001100011000110001100011000110001100011000", numOps = 50) == 1

Answer: assert Solution().minLength(s = "10101010101010101010", numOps = 15) == 1

Answer: assert Solution().minLength(s = "1111000011110000", numOps = 4) == 2

Answer: assert Solution().minLength(s = "0100101010010101001010100101010010101001010100101010", numOps = 20) == 2

Answer: assert Solution().minLength(s = "11111111111111111111", numOps = 10) == 1

Answer: assert Solution().minLength(s = "1111111111", numOps = 10) == 1

Answer: assert Solution().minLength(s = "000000000011111111", numOps = 9) == 1

Answer: assert Solution().minLength(s = "010101010101010101010101010101010101010101010101", numOps = 100) == 1

Answer: assert Solution().minLength(s = "10101010101010101010101010101010101010101010101010", numOps = 25) == 1

Answer: assert Solution().minLength(s = "101010101010101010101010101010101010101010101010", numOps = 0) == 1

Answer: assert Solution().minLength(s = "001100110011001100110011001100110011001100110011", numOps = 15) == 2

Answer: assert Solution().minLength(s = "1100110011", numOps = 4) == 2

Answer: assert Solution().minLength(s = "1010101010101010101010101010101010101010101010101010", numOps = 100) == 1

Answer: assert Solution().minLength(s = "00110011001100110011", numOps = 8) == 2

Answer: assert Solution().minLength(s = "000001111110000011111100000", numOps = 12) == 2

Answer: assert Solution().minLength(s = "11001100110011", numOps = 6) == 2

Answer: assert Solution().minLength(s = "111001110011100111001110011100111001110011100111", numOps = 45) == 1

Answer: assert Solution().minLength(s = "1111111111111111", numOps = 15) == 1

Answer: assert Solution().minLength(s = "100110011001100110011001100110011001100110011001", numOps = 25) == 1

Answer: assert Solution().minLength(s = "111100001111000011110000111100001111000011110000", numOps = 35) == 1

Answer: assert Solution().minLength(s = "0000000000000000000000000000000000000000000000000000000000000000", numOps = 100) == 1

Answer: assert Solution().minLength(s = "1001100110011001100110011", numOps = 10) == 2

Answer: assert Solution().minLength(s = "101010101010101010101010", numOps = 12) == 1

Answer: assert Solution().minLength(s = "1111111111111111111111111111111111111111111111111111", numOps = 30) == 1

Answer: assert Solution().minLength(s = "011011011011011011011011011011011011011011011011", numOps = 50) == 1

Answer: assert Solution().minLength(s = "1100110011001100", numOps = 6) == 2

Answer: assert Solution().minLength(s = "101010101010101010101010", numOps = 7) == 1

Answer: assert Solution().minLength(s = "1001001001001001001001", numOps = 10) == 2

Answer: assert Solution().minLength(s = "11111111111111111111", numOps = 5) == 3

Answer: assert Solution().minLength(s = "111000111000111000", numOps = 5) == 3

Answer: assert Solution().minLength(s = "00110011001100110011", numOps = 10) == 1

Answer: assert Solution().minLength(s = "001100110011", numOps = 5) == 2

Answer: assert Solution().minLength(s = "00000111110000011111000001111100000111110000011111", numOps = 20) == 1

Answer: assert Solution().minLength(s = "1111111111111111111111111111111111111111111111", numOps = 15) == 2

Answer: assert Solution().minLength(s = "000000000011111111110000000000111111111100000000", numOps = 40) == 1

Answer: assert Solution().minLength(s = "0101010101010101010101010101", numOps = 10) == 1

Answer: assert Solution().minLength(s = "000110001100011", numOps = 5) == 2

Answer: assert Solution().minLength(s = "111000111000111", numOps = 6) == 1

Answer: assert Solution().minLength(s = "000011110000111100001111", numOps = 10) == 2

Answer: assert Solution().minLength(s = "1010101010101010101010101010101010101010101010", numOps = 25) == 1

Answer: assert Solution().minLength(s = "00000000000000000000000000000000000000000000000000", numOps = 50) == 1

Answer: assert Solution().minLength(s = "1111000000", numOps = 3) == 2

Answer: assert Solution().minLength(s = "1010101010101010101010101010", numOps = 15) == 1

Answer: assert Solution().minLength(s = "10101010101010101010101010101010", numOps = 20) == 1

Answer: assert Solution().minLength(s = "100100100100100100100100100100100100100100100100", numOps = 55) == 1

Answer: assert Solution().minLength(s = "00000000000000000000000000000000000000000000000000", numOps = 30) == 1

Answer: assert Solution().minLength(s = "111111000000111111000000111111", numOps = 8) == 3

Answer: assert Solution().minLength(s = "1111111111000000000000000000000000000000000000000000", numOps = 30) == 1

Answer: assert Solution().minLength(s = "111111111111111111111111", numOps = 25) == 1

Answer: assert Solution().minLength(s = "10010010010010010010", numOps = 3) == 2

Answer: assert Solution().minLength(s = "000000000000000000000000000000000000000000000000", numOps = 25) == 1

Answer: assert Solution().minLength(s = "010101010101010101010101010101010101010101010101", numOps = 10) == 1

Answer: assert Solution().minLength(s = "000111000111000111", numOps = 6) == 1

Answer: assert Solution().minLength(s = "110001110011", numOps = 4) == 2

Answer: assert Solution().minLength(s = "110011001100110011001100", numOps = 10) == 2

Answer: assert Solution().minLength(s = "111111111111111111110000000000000000000000000000", numOps = 50) == 1

Answer: assert Solution().minLength(s = "00001111000011110000", numOps = 6) == 2

Answer: assert Solution().minLength(s = "11110000111100001111000011110000", numOps = 20) == 1

Answer: assert Solution().minLength(s = "0000000000000000000000000000", numOps = 10) == 2

Answer: assert Solution().minLength(s = "111111111111111111111111", numOps = 15) == 1

Answer: assert Solution().minLength(s = "111111111111111111111111111111111111111111111111", numOps = 20) == 2

Answer: assert Solution().minLength(s = "000000000000000000001111111111111111111111111111", numOps = 50) == 1

Answer: assert Solution().minLength(s = "000000000000000000000000000000000000000000000000000000000000", numOps = 50) == 1

Answer: assert Solution().minLength(s = "11001100110011001100", numOps = 4) == 2

Answer: assert Solution().minLength(s = "111111111100000000001111111111110000000000001111", numOps = 60) == 1

Answer: assert Solution().minLength(s = "11000110001100011000", numOps = 15) == 1

import random
import functools
import collections
import string
import math
import datetime

from typing import *
from functools import *
from collections import *
from itertools import *
from heapq import *
from bisect import *
from string import *
from operator import *
from math import *

inf = float('inf')

class ListNode:
    def __init__(self, val=0, next=None):
        self.val = val
        self.next = next

def list_node(values: list):
    if not values:
        return None
    head = ListNode(values[0])
    p = head
    for val in values[1:]:
        node = ListNode(val)
        p.next = node
        p = node
    return head

def is_same_list(p1, p2):
    if p1 is None and p2 is None:
        return True
    if not p1 or not p2:
        return False
    return p1.val == p2.val and is_same_list(p1.next, p2.next)

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def tree_node(values: list):
    if not values:
        return None
    root = TreeNode(values[0])
    i = 1
    queue = deque()
    queue.append(root)
    while queue:
        node = queue.popleft()
        if i < len(values) and values[i] is not None:
            node.left = TreeNode(values[i])
            queue.append(node.left)
        i += 1
        if i < len(values) and values[i] is not None:
            node.right = TreeNode(values[i])
            queue.append(node.right)
        i += 1
    return root

def is_same_tree(p, q):
    if not p and not q:
        return True
    elif not p or not q:
        return False
    elif p.val != q.val:
        return False
    else:
        return is_same_tree(p.left, q.left) and is_same_tree(p.right, q.right)

class Solution:
    def minLength(self, s: str, numOps: int) -> int:
        def check(m: int) -> bool:
            cnt = 0
            if m == 1:
                t = "01"
                cnt = sum(c == t[i & 1] for i, c in enumerate(s))
                cnt = min(cnt, n - cnt)
            else:
                k = 0
                for i, c in enumerate(s):
                    k += 1
                    if i == len(s) - 1 or c != s[i + 1]:
                        cnt += k // (m + 1)
                        k = 0
            return cnt <= numOps

        n = len(s)
        return bisect_left(range(n), True, lo=1, key=check)


You are an expert Python programmer. Write correct, efficient Python 3 code that solves the problem and passes all tests. Output the code in the following format: ```python
<code>
```

PROBLEM:
You are given two integer arrays, nums1 and nums2, of the same length.
An index i is considered matching if nums1[i] == nums2[i].
Return the maximum number of matching indices after performing any number of right shifts on nums1.
A right shift is defined as shifting the element at index i to index (i + 1) % n, for all indices.

Example 1:

Input: nums1 = [3,1,2,3,1,2], nums2 = [1,2,3,1,2,3]
Output: 6
Explanation:
If we right shift nums1 2 times, it becomes [1, 2, 3, 1, 2, 3]. Every index matches, so the output is 6.

Example 2:

Input: nums1 = [1,4,2,5,3,1], nums2 = [2,3,1,2,4,6]
Output: 3
Explanation:
If we right shift nums1 3 times, it becomes [5, 3, 1, 1, 4, 2]. Indices 1, 2, and 4 match, so the output is 3.

Constraints:

nums1.length == nums2.length
1 <= nums1.length, nums2.length <= 3000
1 <= nums1[i], nums2[i] <= 109

Your solution to the problem should be a method of the class Solution called maximumMatchingIndices and should pass all tests. Use the following starter code:
```python
class Solution:
    def maximumMatchingIndices(self, nums1: List[int], nums2: List[int]) -> int:
        # Add your code here
```

SOLUTION:

Question:
PROBLEM:
You are given two integer arrays, nums1 and nums2, of the same length.
An index i is considered matching if nums1[i] == nums2[i].
Return the maximum number of matching indices after performing any number of right shifts on nums1.
A right shift is defined as shifting the element at index i to index (i + 1) % n, for all indices.

Example 1:

Input: nums1 = [3,1,2,3,1,2], nums2 = [1,2,3,1,2,3]
Output: 6
Explanation:
If we right shift nums1 2 times, it becomes [1, 2, 3, 1, 2, 3]. Every index matches, so the output is 6.

Example 2:

Input: nums1 = [1,4,2,5,3,1], nums2 = [2,3,1,2,4,6]
Output: 3
Explanation:
If we right shift nums1 3 times, it becomes [5, 3, 1, 1, 4, 2]. Indices 1, 2, and 4 match, so the output is 3.

Constraints:

nums1.length == nums2.length
1 <= nums1.length, nums2.length <= 3000
1 <= nums1[i], nums2[i] <= 109

Your solution to the problem should be a method of the class Solution called maximumMatchingIndices and should pass all tests. Use the following starter code:
```python
class Solution:
    def maximumMatchingIndices(self, nums1: List[int], nums2: List[int]) -> int:
        # Add your code here
```

SOLUTION:

assert Solution().maximumMatchingIndices(nums1 = [1,3,5,7,9], nums2 = [2,4,6,8,0]) == 0

assert Solution().maximumMatchingIndices(nums1 = [1,4,2,5,3,1], nums2 = [2,3,1,2,4,6]) == 3

assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5], nums2 = [5,1,2,3,4]) == 5

assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1], nums2 = [1,1,1,1]) == 4

assert Solution().maximumMatchingIndices(nums1 = [7,8,9,7,8,9], nums2 = [8,9,7,8,9,7]) == 6

assert Solution().maximumMatchingIndices(nums1 = [1,2,3], nums2 = [3,2,1]) == 1

assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1,1,1], nums2 = [1,1,1,1,1,1]) == 6

assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5,5], nums2 = [5,5,5,5,5]) == 5

assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5], nums2 = [5,4,3,2,1]) == 1

assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5], nums2 = [5,5,5,5]) == 4

assert Solution().maximumMatchingIndices(nums1 = [9,8,7,6,5,4], nums2 = [4,9,8,7,6,5]) == 6

assert Solution().maximumMatchingIndices(nums1 = [1,2,3], nums2 = [3,1,2]) == 3

assert Solution().maximumMatchingIndices(nums1 = [7,8,9], nums2 = [9,7,8]) == 3

assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6], nums2 = [6,5,4,3,2,1]) == 2

assert Solution().maximumMatchingIndices(nums1 = [7,8,9,10,11], nums2 = [7,8,9,10,11]) == 5

assert Solution().maximumMatchingIndices(nums1 = [3,1,2,3,1,2], nums2 = [1,2,3,1,2,3]) == 6

assert Solution().maximumMatchingIndices(nums1 = [7,8,9,10,11], nums2 = [11,7,8,9,10]) == 5

assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1,1], nums2 = [1,1,1,1,1]) == 5

assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5,5], nums2 = [5,5,5,5,6]) == 4

assert Solution().maximumMatchingIndices(nums1 = [100,200,300], nums2 = [300,100,200]) == 3

assert Solution().maximumMatchingIndices(nums1 = [1, 2, 2, 3, 3, 4, 4], nums2 = [4, 4, 1, 2, 2, 3, 3]) == 7

assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7], nums2 = [7, 6, 5, 4, 3, 2, 1]) == 1

assert Solution().maximumMatchingIndices(nums1 = [5, 5, 5, 5, 5, 5], nums2 = [5, 5, 5, 5, 5, 5]) == 6

assert Solution().maximumMatchingIndices(nums1 = [1, 1, 1, 1, 1, 1, 1], nums2 = [1, 1, 1, 1, 1, 1, 1]) == 7

assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1,1,1,1], nums2 = [1,1,1,1,1,1,1]) == 7

Answer: assert Solution().maximumMatchingIndices(nums1 = [10,20,30,40,50], nums2 = [50,10,20,30,40]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10], nums2 = [10,1,2,3,4,5,6,7,8,9]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,3,5,7,9], nums2 = [2,4,6,8,0]) == 0

Answer: assert Solution().maximumMatchingIndices(nums1 = [10,20,30,40,50], nums2 = [15,25,35,45,55]) == 0

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,4,2,5,3,1], nums2 = [2,3,1,2,4,6]) == 3

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5], nums2 = [5,1,2,3,4]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1], nums2 = [1,1,1,1]) == 4

Answer: assert Solution().maximumMatchingIndices(nums1 = [7,8,9,7,8,9], nums2 = [8,9,7,8,9,7]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [100,200,300,400,500], nums2 = [100,200,300,400,500]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3], nums2 = [3,2,1]) == 1

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1,1,1], nums2 = [1,1,1,1,1,1]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5,5], nums2 = [5,5,5,5,5]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5], nums2 = [5,4,3,2,1]) == 1

Answer: assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5], nums2 = [5,5,5,5]) == 4

Answer: assert Solution().maximumMatchingIndices(nums1 = [9,8,7,6,5,4], nums2 = [4,9,8,7,6,5]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3], nums2 = [3,1,2]) == 3

Answer: assert Solution().maximumMatchingIndices(nums1 = [7,8,9], nums2 = [9,7,8]) == 3

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6], nums2 = [6,5,4,3,2,1]) == 2

Answer: assert Solution().maximumMatchingIndices(nums1 = [7,8,9,10,11], nums2 = [7,8,9,10,11]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [3,1,2,3,1,2], nums2 = [1,2,3,1,2,3]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [7,8,9,10,11], nums2 = [11,7,8,9,10]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1,1], nums2 = [1,1,1,1,1]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5,5], nums2 = [5,5,5,5,6]) == 4

Answer: assert Solution().maximumMatchingIndices(nums1 = [100,200,300], nums2 = [300,100,200]) == 3

Answer: assert Solution().maximumMatchingIndices(nums1 = [10, 20, 30, 40, 50], nums2 = [50, 10, 20, 30, 40]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10], nums2 = [1,2,3,4,5,6,7,8,9,10,2,3,4,5,6,7,8,9,10]) == 19

Answer: assert Solution().maximumMatchingIndices(nums1 = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], nums2 = [4, 3, 2, 1, 10, 9, 8, 7, 6, 5]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [999999999, 999999998, 999999997, 999999996, 999999995], nums2 = [999999995, 999999999, 999999998, 999999997, 999999996]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1000000000, 2000000000, 3000000000, 1000000000, 2000000000, 3000000000], nums2 = [3000000000, 1000000000, 2000000000, 3000000000, 1000000000, 2000000000]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 2, 3, 3, 4, 4], nums2 = [4, 4, 1, 2, 2, 3, 3]) == 7

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], nums2 = [6, 7, 8, 9, 10, 1, 2, 3, 4, 5]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], nums2 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5], nums2 = [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], nums2 = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [123,456,789,123,456,789,123,456,789,123], nums2 = [456,789,123,456,789,123,456,789,123,456]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], nums2 = [11,12,13,14,15,16,17,18,19,20,1,2,3,4,5,6,7,8,9,10]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7], nums2 = [7, 6, 5, 4, 3, 2, 1]) == 1

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 3, 5, 7, 9, 11, 13, 15], nums2 = [9, 11, 13, 15, 1, 3, 5, 7]) == 8

Answer: assert Solution().maximumMatchingIndices(nums1 = [7,8,9,10,11,12,13], nums2 = [13,7,8,9,10,11,12]) == 7

Answer: assert Solution().maximumMatchingIndices(nums1 = [100,200,300,400,500,600,700,800,900,1000], nums2 = [100,200,300,400,500,600,700,800,900,1000]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [100,200,300,400,500], nums2 = [500,100,200,300,400]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], nums2 = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1]) == 2

Answer: assert Solution().maximumMatchingIndices(nums1 = [4, 8, 15, 16, 23, 42, 4, 8, 15, 16, 23, 42], nums2 = [4, 8, 15, 16, 23, 42, 1, 2, 3, 4, 5, 6]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [10,20,30,10,20,30,10,20,30], nums2 = [20,30,10,20,30,10,20,30,10]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [5, 5, 5, 5, 5, 5, 5, 5, 5], nums2 = [5, 5, 5, 5, 5, 5, 5, 5, 5]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [1000000000,2000000000,3000000000], nums2 = [3000000000,1000000000,2000000000]) == 3

Answer: assert Solution().maximumMatchingIndices(nums1 = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], nums2 = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], nums2 = [8, 9, 10, 11, 12, 13, 14, 15, 1, 2, 3, 4, 5, 6, 7]) == 15

Answer: assert Solution().maximumMatchingIndices(nums1 = [10,9,8,7,6,5,4,3,2,1], nums2 = [5,4,3,2,1,10,9,8,7,6]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [10,20,30,40,50,60,70,80,90,100], nums2 = [90,100,10,20,30,40,50,60,70,80]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [100,200,300,400,500,600,700,800,900,1000], nums2 = [1000,100,200,300,400,500,600,700,800,900]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [7,7,7,7,7,7,7,7,7,7], nums2 = [7,7,7,7,7,7,7,7,7,7]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,3,5,7,9,11,13,15,17,19], nums2 = [3,5,7,9,11,13,15,17,19,1]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [5, 5, 5, 5, 5, 5], nums2 = [5, 5, 5, 5, 5, 5]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [5,4,3,2,1,5,4,3,2,1], nums2 = [1,5,4,3,2,1,5,4,3,2]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 3, 5, 7, 9, 11, 13, 15, 17], nums2 = [17, 1, 3, 5, 7, 9, 11, 13, 15]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42], nums2 = [42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42, 42]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], nums2 = [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]) == 1

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], nums2 = [19, 1, 3, 5, 7, 9, 11, 13, 15, 17]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,3,5,7,9,11,13,15,17,19], nums2 = [2,4,6,8,10,12,14,16,18,20]) == 0

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 1, 2, 2, 3, 3, 4, 4], nums2 = [3, 3, 4, 4, 1, 1, 2, 2]) == 8

Answer: assert Solution().maximumMatchingIndices(nums1 = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50], nums2 = [15, 20, 25, 30, 35, 40, 45, 50, 5, 10]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10], nums2 = [2,3,4,5,6,7,8,9,10,1]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [4,5,6,4,5,6,4,5,6], nums2 = [5,6,4,5,6,4,5,6,4]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3], nums2 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) == 1

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,3,5,7,9,11,13,15,17,19], nums2 = [19,1,3,5,7,9,11,13,15,17]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [100, 200, 300, 400, 500], nums2 = [300, 400, 500, 100, 200]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 3, 5, 7, 9, 11, 13, 15], nums2 = [15, 1, 3, 5, 7, 9, 11, 13]) == 8

Answer: assert Solution().maximumMatchingIndices(nums1 = [7,8,9,1,2,3,4,5,6], nums2 = [4,5,6,7,8,9,1,2,3]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10], nums2 = [10,9,8,7,6,5,4,3,2,1]) == 2

Answer: assert Solution().maximumMatchingIndices(nums1 = [9,8,7,6,5,4,3,2,1], nums2 = [7,6,5,4,3,2,1,9,8]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [5, 4, 3, 2, 1, 5, 4, 3, 2, 1], nums2 = [1, 2, 3, 4, 5, 1, 2, 3, 4, 5]) == 2

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,1,2,3,4,5,1,2,3,4,5], nums2 = [2,3,4,5,1,2,3,4,5,1,2,3,4,5,1]) == 15

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], nums2 = [29, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27]) == 15

Answer: assert Solution().maximumMatchingIndices(nums1 = [5,3,4,2,1,3,4,5,2,1], nums2 = [1,3,4,5,2,1,5,3,4,2]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [9,8,7,6,5,4,3,2,1], nums2 = [1,2,3,4,5,6,7,8,9]) == 1

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 1, 2, 1, 2, 1, 2, 1, 2], nums2 = [2, 1, 2, 1, 2, 1, 2, 1, 2, 1]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 1, 1, 1, 1, 1, 1], nums2 = [1, 1, 1, 1, 1, 1, 1]) == 7

Answer: assert Solution().maximumMatchingIndices(nums1 = [10, 20, 30, 40, 50, 10, 20], nums2 = [20, 30, 40, 50, 10, 20, 30]) == 6

Answer: assert Solution().maximumMatchingIndices(nums1 = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], nums2 = [100, 90, 80, 70, 60, 50, 40, 30, 20, 10]) == 2

Answer: assert Solution().maximumMatchingIndices(nums1 = [1000000000, 999999999, 888888888, 777777777], nums2 = [999999999, 888888888, 777777777, 1000000000]) == 4

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1,1,1,1], nums2 = [1,1,1,1,1,1,1]) == 7

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 1, 2, 3, 4, 5], nums2 = [5, 1, 2, 3, 4, 5, 1, 2, 3, 4]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9], nums2 = [9, 1, 2, 3, 4, 5, 6, 7, 8]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], nums2 = [3,4,5,6,7,8,9,10,11,12,13,14,15,1,2]) == 15

Answer: assert Solution().maximumMatchingIndices(nums1 = [5,5,5,5,5,5,5,5,5,5], nums2 = [5,5,5,5,5,5,5,5,5,5]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], nums2 = [15,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 15

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], nums2 = [20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,1,1,1,1,1,1,1,1,1], nums2 = [1,1,1,1,1,1,1,1,1,1]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,1,2,3,1,2,3,1], nums2 = [2,3,1,2,3,1,2,3,1,2]) == 9

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,1,2,2,3,3,4,4,5,5], nums2 = [2,2,1,1,4,4,3,3,5,5]) == 4

Answer: assert Solution().maximumMatchingIndices(nums1 = [3,1,4,1,5,9,2,6,5,3,5,9], nums2 = [9,2,6,5,3,5,9,3,1,4,1,5]) == 12

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], nums2 = [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], nums2 = [20,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [3,1,4,1,5,9,2,6,5,3,5,9,7,9,3,2,3,8,4,6], nums2 = [2,7,1,8,2,8,1,8,2,8,4,5,9,0,4,5,2,3,5,3]) == 5

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], nums2 = [2, 3, 4, 5, 6, 7, 8, 9, 10, 1]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [7, 8, 9, 10, 11, 12, 13], nums2 = [10, 11, 12, 13, 7, 8, 9]) == 7

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 1, 1, 1, 1, 2, 2, 2, 2, 2], nums2 = [2, 2, 2, 2, 2, 1, 1, 1, 1, 1]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,3,4,5,6,7,8,9,10], nums2 = [6,7,8,9,10,1,2,3,4,5]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], nums2 = [12, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]) == 12

Answer: assert Solution().maximumMatchingIndices(nums1 = [10,20,30,40,50,60,70,80,90,100], nums2 = [60,70,80,90,100,10,20,30,40,50]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], nums2 = [1000, 100, 200, 300, 400, 500, 600, 700, 800, 900]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,3,5,7,9,11,13,15,17,19], nums2 = [11,13,15,17,19,1,3,5,7,9]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], nums2 = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30], nums2 = [26, 27, 28, 29, 30, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25]) == 30

Answer: assert Solution().maximumMatchingIndices(nums1 = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], nums2 = [600, 700, 800, 900, 1000, 100, 200, 300, 400, 500]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [10,20,30,40,50,60,70,80,90,100], nums2 = [40,50,60,70,80,90,100,10,20,30]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,1,2,2,3,3,4,4,5,5], nums2 = [5,5,1,1,2,2,3,3,4,4]) == 10

Answer: assert Solution().maximumMatchingIndices(nums1 = [7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1], nums2 = [1, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2]) == 14

Answer: assert Solution().maximumMatchingIndices(nums1 = [3,3,3,3,1,1,1,1,2,2,2,2], nums2 = [2,2,2,2,3,3,3,3,1,1,1,1]) == 12

Answer: assert Solution().maximumMatchingIndices(nums1 = [1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2], nums2 = [2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1,2,1]) == 20

Answer: assert Solution().maximumMatchingIndices(nums1 = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], nums2 = [4, 5, 6, 7, 8, 9, 10, 1, 2, 3]) == 10

import random
import functools
import collections
import string
import math
import datetime

from typing import *
from functools import *
from collections import *
from itertools import *
from heapq import *
from bisect import *
from string import *
from operator import *
from math import *

inf = float('inf')

class ListNode:
    def __init__(self, val=0, next=None):
        self.val = val
        self.next = next

def list_node(values: list):
    if not values:
        return None
    head = ListNode(values[0])
    p = head
    for val in values[1:]:
        node = ListNode(val)
        p.next = node
        p = node
    return head

def is_same_list(p1, p2):
    if p1 is None and p2 is None:
        return True
    if not p1 or not p2:
        return False
    return p1.val == p2.val and is_same_list(p1.next, p2.next)

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def tree_node(values: list):
    if not values:
        return None
    root = TreeNode(values[0])
    i = 1
    queue = deque()
    queue.append(root)
    while queue:
        node = queue.popleft()
        if i < len(values) and values[i] is not None:
            node.left = TreeNode(values[i])
            queue.append(node.left)
        i += 1
        if i < len(values) and values[i] is not None:
            node.right = TreeNode(values[i])
            queue.append(node.right)
        i += 1
    return root

def is_same_tree(p, q):
    if not p and not q:
        return True
    elif not p or not q:
        return False
    elif p.val != q.val:
        return False
    else:
        return is_same_tree(p.left, q.left) and is_same_tree(p.right, q.right)

class Solution:
    def maximumMatchingIndices(self, nums1: List[int], nums2: List[int]) -> int:
        n = len(nums1)
        ans = 0
        for k in range(n):
            t = sum(nums1[(i + k) % n] == x for i, x in enumerate(nums2))
            ans = max(ans, t)
        return ans


You are an expert Python programmer. Write correct, efficient Python 3 code that solves the problem and passes all tests. Output the code in the following format: ```python
<code>
```

PROBLEM:
You are given a string word, and an integer numFriends.
Alice is organizing a game for her numFriends friends. There are multiple rounds in the game, where in each round:

word is split into numFriends non-empty strings, such that no previous round has had the exact same split.
All the split words are put into a box.

Find the lexicographically largest string from the box after all the rounds are finished.

Example 1:

Input: word = "dbca", numFriends = 2
Output: "dbc"
Explanation:
All possible splits are:

"d" and "bca".
"db" and "ca".
"dbc" and "a".

Example 2:

Input: word = "gggg", numFriends = 4
Output: "g"
Explanation:
The only possible split is: "g", "g", "g", and "g".

Constraints:

1 <= word.length <= 5 * 103
word consists only of lowercase English letters.
1 <= numFriends <= word.length

Your solution to the problem should be a method of the class Solution called answerString and should pass all tests. Use the following starter code:
```python
class Solution:
    def answerString(self, word: str, numFriends: int) -> str:
        # Add your code here
```

SOLUTION:

Question:
PROBLEM:
You are given a string word, and an integer numFriends.
Alice is organizing a game for her numFriends friends. There are multiple rounds in the game, where in each round:

word is split into numFriends non-empty strings, such that no previous round has had the exact same split.
All the split words are put into a box.

Find the lexicographically largest string from the box after all the rounds are finished.

Example 1:

Input: word = "dbca", numFriends = 2
Output: "dbc"
Explanation:
All possible splits are:

"d" and "bca".
"db" and "ca".
"dbc" and "a".

Example 2:

Input: word = "gggg", numFriends = 4
Output: "g"
Explanation:
The only possible split is: "g", "g", "g", and "g".

Constraints:

1 <= word.length <= 5 * 103
word consists only of lowercase English letters.
1 <= numFriends <= word.length

Your solution to the problem should be a method of the class Solution called answerString and should pass all tests. Use the following starter code:
```python
class Solution:
    def answerString(self, word: str, numFriends: int) -> str:
        # Add your code here
```

SOLUTION:

assert Solution().answerString(word = "aabbcc", numFriends = 2) == "cc"

assert Solution().answerString(word = "zxy", numFriends = 1) == "zxy"

assert Solution().answerString(word = "zyxwvutsrqponmlkjihgfedcba", numFriends = 5) == "zyxwvutsrqponmlkjihgfe"

assert Solution().answerString(word = "aabbccddeeff", numFriends = 6) == "ff"

assert Solution().answerString(word = "gggg", numFriends = 4) == "g"

assert Solution().answerString(word = "aabbcc", numFriends = 6) == "c"

assert Solution().answerString(word = "abcdefg", numFriends = 7) == "g"

assert Solution().answerString(word = "abcdef", numFriends = 3) == "f"

assert Solution().answerString(word = "banana", numFriends = 3) == "nana"

assert Solution().answerString(word = "zzzzzz", numFriends = 2) == "zzzzz"

assert Solution().answerString(word = "racecar", numFriends = 5) == "rac"

assert Solution().answerString(word = "abcde", numFriends = 3) == "e"

assert Solution().answerString(word = "zyxwvut", numFriends = 3) == "zyxwv"

assert Solution().answerString(word = "zxy", numFriends = 3) == "z"

assert Solution().answerString(word = "abcde", numFriends = 2) == "e"

assert Solution().answerString(word = "dbca", numFriends = 2) == "dbc"

assert Solution().answerString(word = "abcdef", numFriends = 2) == "f"

assert Solution().answerString(word = "aabbcc", numFriends = 3) == "cc"

assert Solution().answerString(word = "xyzzxyzzyx", numFriends = 5) == "zzyx"

assert Solution().answerString(word = "aaaaabbbbccccdddd", numFriends = 10) == "dddd"

assert Solution().answerString(word = "level", numFriends = 3) == "vel"

assert Solution().answerString(word = "aaaaabbbbccccddddeeeee", numFriends = 5) == "eeeee"

assert Solution().answerString(word = "leetcodeisfun", numFriends = 5) == "un"

assert Solution().answerString(word = "abcdabcdabcdabcd", numFriends = 4) == "dabcdabcdabcd"

assert Solution().answerString(word = "abcdefghij", numFriends = 5) == "j"

assert Solution().answerString(word = "amazingrace", numFriends = 3) == "zingrace"

assert Solution().answerString(word = "racecar", numFriends = 2) == "raceca"

assert Solution().answerString(word = "zzzzyyyyxxxwwwwvvvuuutttsssrrrqqqppoonn", numFriends = 10) == "zzzzyyyyxxxwwwwvvvuuutttsssrrr"

assert Solution().answerString(word = "abcdedcba", numFriends = 5) == "edcba"

assert Solution().answerString(word = "mamamamamamamama", numFriends = 12) == "mamam"

assert Solution().answerString(word = "bananabananabanana", numFriends = 6) == "nanabananaban"

assert Solution().answerString(word = "hello", numFriends = 1) == "hello"

assert Solution().answerString(word = "abcdefghij", numFriends = 10) == "j"

assert Solution().answerString(word = "abcabcabcabcabcabcabcabcabcabc", numFriends = 9) == "cabcabcabcabcabcabcabc"

assert Solution().answerString(word = "zzzzzzyxwvutsrqponmlkjihgfedcba", numFriends = 3) == "zzzzzzyxwvutsrqponmlkjihgfedc"

assert Solution().answerString(word = "abracadabra", numFriends = 5) == "racadab"

assert Solution().answerString(word = "pneumonoultramicroscopicsilicovolcanoconiosis", numFriends = 10) == "volcanoconiosis"

assert Solution().answerString(word = "mississippi", numFriends = 4) == "ssissipp"

assert Solution().answerString(word = "abcdabcdabcdabcdabcdabcdabcdabcd", numFriends = 4) == "dabcdabcdabcdabcdabcdabcdabcd"

assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 5) == "z"

assert Solution().answerString(word = "zyxwvutsrqponmlkjihgfedcba", numFriends = 10) == "zyxwvutsrqponmlkj"

assert Solution().answerString(word = "abcdabcdaaaa", numFriends = 5) == "dabcdaaa"

assert Solution().answerString(word = "deified", numFriends = 2) == "ified"

assert Solution().answerString(word = "madam", numFriends = 3) == "mad"

assert Solution().answerString(word = "abcdabcdabcdabcd", numFriends = 5) == "dabcdabcdabc"

assert Solution().answerString(word = "oneonetwoonethree", numFriends = 4) == "woonethree"

assert Solution().answerString(word = "aaaabbbbcccc", numFriends = 6) == "cccc"

assert Solution().answerString(word = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzz", numFriends = 3) == "zzzzzzzzzzzzzzzzzzzzzzzzzzz"

assert Solution().answerString(word = "zyxwvutsrqponmlkjihgfedcba", numFriends = 3) == "zyxwvutsrqponmlkjihgfedc"

assert Solution().answerString(word = "zzzzzzzzzzzzzzzzzzzz", numFriends = 10) == "zzzzzzzzzzz"

assert Solution().answerString(word = "hellohellohello", numFriends = 3) == "ohellohello"

assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 26) == "z"

assert Solution().answerString(word = "leetcode", numFriends = 3) == "tcode"

assert Solution().answerString(word = "abacabadabacaba", numFriends = 7) == "dabacaba"

assert Solution().answerString(word = "aaaaaaaaaaa", numFriends = 3) == "aaaaaaaaa"

assert Solution().answerString(word = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", numFriends = 10) == "zz"

assert Solution().answerString(word = "xyzxyzxyz", numFriends = 3) == "zxyzxyz"

assert Solution().answerString(word = "zzzzyyyy", numFriends = 4) == "zzzzy"

assert Solution().answerString(word = "aabbccddeeffgghh", numFriends = 8) == "hh"

assert Solution().answerString(word = "elephant", numFriends = 2) == "t"

assert Solution().answerString(word = "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", numFriends = 100) == ""

assert Solution().answerString(word = "xyzxyzxyzxyz", numFriends = 4) == "zxyzxyzxy"

assert Solution().answerString(word = "zyxzyxzyxzyxzyxzyxzyx", numFriends = 5) == "zyxzyxzyxzyxzyxzy"

assert Solution().answerString(word = "zzzzzzzz", numFriends = 8) == "z"

assert Solution().answerString(word = "lemonadelemonade", numFriends = 6) == "onadelemona"

assert Solution().answerString(word = "aaaabbbbccccddddeeeeffff", numFriends = 10) == "ffff"

assert Solution().answerString(word = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", numFriends = 26) == "zz"

assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 1) == "abcdefghijklmnopqrstuvwxyz"

assert Solution().answerString(word = "racecar", numFriends = 3) == "racec"

assert Solution().answerString(word = "abcabcabcabc", numFriends = 8) == "cabca"

assert Solution().answerString(word = "mississippi", numFriends = 3) == "ssissippi"

assert Solution().answerString(word = "xyzzzxyzzzxyzzz", numFriends = 3) == "zzzxyzzzxyzzz"

assert Solution().answerString(word = "qwertyuiopasdfghjklzxcvbnm", numFriends = 5) == "zxcvbnm"

assert Solution().answerString(word = "refer", numFriends = 5) == "r"

assert Solution().answerString(word = "banana", numFriends = 2) == "nana"

assert Solution().answerString(word = "hellohellomyfriend", numFriends = 7) == "yfriend"

assert Solution().answerString(word = "abcabcabcabcabcabc", numFriends = 6) == "cabcabcabcabc"

assert Solution().answerString(word = "pqrsrstuv", numFriends = 4) == "v"

assert Solution().answerString(word = "zzzzzzzzzzzzzzzzzzzz", numFriends = 20) == "z"

assert Solution().answerString(word = "zzzzzzyyyyyy", numFriends = 3) == "zzzzzzyyyy"

assert Solution().answerString(word = "thisisalongword", numFriends = 4) == "word"

assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 10) == "z"

assert Solution().answerString(word = "leetcodecontest", numFriends = 4) == "test"

assert Solution().answerString(word = "mnopqrstu", numFriends = 3) == "u"

assert Solution().answerString(word = "abracadabra", numFriends = 3) == "racadabra"

assert Solution().answerString(word = "rotor", numFriends = 1) == "rotor"

assert Solution().answerString(word = "hello", numFriends = 5) == "o"

assert Solution().answerString(word = "hellothere", numFriends = 2) == "there"

assert Solution().answerString(word = "zyxzyxzyx", numFriends = 4) == "zyxzyx"

assert Solution().answerString(word = "aaaabbbbccccdddd", numFriends = 4) == "dddd"

assert Solution().answerString(word = "aaaabbbbccccdddd", numFriends = 8) == "dddd"

assert Solution().answerString(word = "xyzzyx", numFriends = 3) == "zzyx"

assert Solution().answerString(word = "abacabadabacaba", numFriends = 5) == "dabacaba"

Answer: assert Solution().answerString(word = "aabbcc", numFriends = 2) == "cc"

Answer: assert Solution().answerString(word = "zxy", numFriends = 1) == "zxy"

Answer: assert Solution().answerString(word = "zyxwvutsrqponmlkjihgfedcba", numFriends = 5) == "zyxwvutsrqponmlkjihgfe"

Answer: assert Solution().answerString(word = "aabbccddeeff", numFriends = 6) == "ff"

Answer: assert Solution().answerString(word = "gggg", numFriends = 4) == "g"

Answer: assert Solution().answerString(word = "aabbcc", numFriends = 6) == "c"

Answer: assert Solution().answerString(word = "abcdefg", numFriends = 7) == "g"

Answer: assert Solution().answerString(word = "abcdef", numFriends = 3) == "f"

Answer: assert Solution().answerString(word = "banana", numFriends = 3) == "nana"

Answer: assert Solution().answerString(word = "zzzzzz", numFriends = 2) == "zzzzz"

Answer: assert Solution().answerString(word = "racecar", numFriends = 5) == "rac"

Answer: assert Solution().answerString(word = "abcde", numFriends = 3) == "e"

Answer: assert Solution().answerString(word = "zyxwvut", numFriends = 3) == "zyxwv"

Answer: assert Solution().answerString(word = "zxy", numFriends = 3) == "z"

Answer: assert Solution().answerString(word = "abcde", numFriends = 2) == "e"

Answer: assert Solution().answerString(word = "dbca", numFriends = 2) == "dbc"

Answer: assert Solution().answerString(word = "abcdef", numFriends = 2) == "f"

Answer: assert Solution().answerString(word = "aabbcc", numFriends = 3) == "cc"

Answer: assert Solution().answerString(word = "xyzzxyzzyx", numFriends = 5) == "zzyx"

Answer: assert Solution().answerString(word = "aaaaabbbbccccdddd", numFriends = 10) == "dddd"

Answer: assert Solution().answerString(word = "level", numFriends = 3) == "vel"

Answer: assert Solution().answerString(word = "aaaaabbbbccccddddeeeee", numFriends = 5) == "eeeee"

Answer: assert Solution().answerString(word = "leetcodeisfun", numFriends = 5) == "un"

Answer: assert Solution().answerString(word = "abcdabcdabcdabcd", numFriends = 4) == "dabcdabcdabcd"

Answer: assert Solution().answerString(word = "abcdefghij", numFriends = 5) == "j"

Answer: assert Solution().answerString(word = "amazingrace", numFriends = 3) == "zingrace"

Answer: assert Solution().answerString(word = "racecar", numFriends = 2) == "raceca"

Answer: assert Solution().answerString(word = "zzzzyyyyxxxwwwwvvvuuutttsssrrrqqqppoonn", numFriends = 10) == "zzzzyyyyxxxwwwwvvvuuutttsssrrr"

Answer: assert Solution().answerString(word = "abcdedcba", numFriends = 5) == "edcba"

Answer: assert Solution().answerString(word = "mamamamamamamama", numFriends = 12) == "mamam"

Answer: assert Solution().answerString(word = "bananabananabanana", numFriends = 6) == "nanabananaban"

Answer: assert Solution().answerString(word = "hello", numFriends = 1) == "hello"

Answer: assert Solution().answerString(word = "abcdefghij", numFriends = 10) == "j"

Answer: assert Solution().answerString(word = "abcabcabcabcabcabcabcabcabcabc", numFriends = 9) == "cabcabcabcabcabcabcabc"

Answer: assert Solution().answerString(word = "zzzzzzyxwvutsrqponmlkjihgfedcba", numFriends = 3) == "zzzzzzyxwvutsrqponmlkjihgfedc"

Answer: assert Solution().answerString(word = "abracadabra", numFriends = 5) == "racadab"

Answer: assert Solution().answerString(word = "pneumonoultramicroscopicsilicovolcanoconiosis", numFriends = 10) == "volcanoconiosis"

Answer: assert Solution().answerString(word = "mississippi", numFriends = 4) == "ssissipp"

Answer: assert Solution().answerString(word = "abcdabcdabcdabcdabcdabcdabcdabcd", numFriends = 4) == "dabcdabcdabcdabcdabcdabcdabcd"

Answer: assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 5) == "z"

Answer: assert Solution().answerString(word = "zyxwvutsrqponmlkjihgfedcba", numFriends = 10) == "zyxwvutsrqponmlkj"

Answer: assert Solution().answerString(word = "abcdabcdaaaa", numFriends = 5) == "dabcdaaa"

Answer: assert Solution().answerString(word = "deified", numFriends = 2) == "ified"

Answer: assert Solution().answerString(word = "madam", numFriends = 3) == "mad"

Answer: assert Solution().answerString(word = "abcdabcdabcdabcd", numFriends = 5) == "dabcdabcdabc"

Answer: assert Solution().answerString(word = "oneonetwoonethree", numFriends = 4) == "woonethree"

Answer: assert Solution().answerString(word = "aaaabbbbcccc", numFriends = 6) == "cccc"

Answer: assert Solution().answerString(word = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzz", numFriends = 3) == "zzzzzzzzzzzzzzzzzzzzzzzzzzz"

Answer: assert Solution().answerString(word = "zyxwvutsrqponmlkjihgfedcba", numFriends = 3) == "zyxwvutsrqponmlkjihgfedc"

Answer: assert Solution().answerString(word = "zzzzzzzzzzzzzzzzzzzz", numFriends = 10) == "zzzzzzzzzzz"

Answer: assert Solution().answerString(word = "hellohellohello", numFriends = 3) == "ohellohello"

Answer: assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 26) == "z"

Answer: assert Solution().answerString(word = "leetcode", numFriends = 3) == "tcode"

Answer: assert Solution().answerString(word = "abacabadabacaba", numFriends = 7) == "dabacaba"

Answer: assert Solution().answerString(word = "aaaaaaaaaaa", numFriends = 3) == "aaaaaaaaa"

Answer: assert Solution().answerString(word = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", numFriends = 10) == "zz"

Answer: assert Solution().answerString(word = "xyzxyzxyz", numFriends = 3) == "zxyzxyz"

Answer: assert Solution().answerString(word = "zzzzyyyy", numFriends = 4) == "zzzzy"

Answer: assert Solution().answerString(word = "aabbccddeeffgghh", numFriends = 8) == "hh"

Answer: assert Solution().answerString(word = "elephant", numFriends = 2) == "t"

Answer: assert Solution().answerString(word = "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", numFriends = 100) == ""

Answer: assert Solution().answerString(word = "xyzxyzxyzxyz", numFriends = 4) == "zxyzxyzxy"

Answer: assert Solution().answerString(word = "zyxzyxzyxzyxzyxzyxzyx", numFriends = 5) == "zyxzyxzyxzyxzyxzy"

Answer: assert Solution().answerString(word = "zzzzzzzz", numFriends = 8) == "z"

Answer: assert Solution().answerString(word = "lemonadelemonade", numFriends = 6) == "onadelemona"

Answer: assert Solution().answerString(word = "aaaabbbbccccddddeeeeffff", numFriends = 10) == "ffff"

Answer: assert Solution().answerString(word = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", numFriends = 26) == "zz"

Answer: assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 1) == "abcdefghijklmnopqrstuvwxyz"

Answer: assert Solution().answerString(word = "racecar", numFriends = 3) == "racec"

Answer: assert Solution().answerString(word = "abcabcabcabc", numFriends = 8) == "cabca"

Answer: assert Solution().answerString(word = "mississippi", numFriends = 3) == "ssissippi"

Answer: assert Solution().answerString(word = "xyzzzxyzzzxyzzz", numFriends = 3) == "zzzxyzzzxyzzz"

Answer: assert Solution().answerString(word = "qwertyuiopasdfghjklzxcvbnm", numFriends = 5) == "zxcvbnm"

Answer: assert Solution().answerString(word = "refer", numFriends = 5) == "r"

Answer: assert Solution().answerString(word = "banana", numFriends = 2) == "nana"

Answer: assert Solution().answerString(word = "hellohellomyfriend", numFriends = 7) == "yfriend"

Answer: assert Solution().answerString(word = "abcabcabcabcabcabc", numFriends = 6) == "cabcabcabcabc"

Answer: assert Solution().answerString(word = "pqrsrstuv", numFriends = 4) == "v"

Answer: assert Solution().answerString(word = "zzzzzzzzzzzzzzzzzzzz", numFriends = 20) == "z"

Answer: assert Solution().answerString(word = "zzzzzzyyyyyy", numFriends = 3) == "zzzzzzyyyy"

Answer: assert Solution().answerString(word = "thisisalongword", numFriends = 4) == "word"

Answer: assert Solution().answerString(word = "abcdefghijklmnopqrstuvwxyz", numFriends = 10) == "z"

Answer: assert Solution().answerString(word = "leetcodecontest", numFriends = 4) == "test"

Answer: assert Solution().answerString(word = "mnopqrstu", numFriends = 3) == "u"

Answer: assert Solution().answerString(word = "abracadabra", numFriends = 3) == "racadabra"

Answer: assert Solution().answerString(word = "rotor", numFriends = 1) == "rotor"

Answer: assert Solution().answerString(word = "hello", numFriends = 5) == "o"

Answer: assert Solution().answerString(word = "hellothere", numFriends = 2) == "there"

Answer: assert Solution().answerString(word = "zyxzyxzyx", numFriends = 4) == "zyxzyx"

Answer: assert Solution().answerString(word = "aaaabbbbccccdddd", numFriends = 4) == "dddd"

Answer: assert Solution().answerString(word = "aaaabbbbccccdddd", numFriends = 8) == "dddd"

Answer: assert Solution().answerString(word = "xyzzyx", numFriends = 3) == "zzyx"

Answer: assert Solution().answerString(word = "abacabadabacaba", numFriends = 5) == "dabacaba"

import random
import functools
import collections
import string
import math
import datetime

from typing import *
from functools import *
from collections import *
from itertools import *
from heapq import *
from bisect import *
from string import *
from operator import *
from math import *

inf = float('inf')

class ListNode:
    def __init__(self, val=0, next=None):
        self.val = val
        self.next = next

def list_node(values: list):
    if not values:
        return None
    head = ListNode(values[0])
    p = head
    for val in values[1:]:
        node = ListNode(val)
        p.next = node
        p = node
    return head

def is_same_list(p1, p2):
    if p1 is None and p2 is None:
        return True
    if not p1 or not p2:
        return False
    return p1.val == p2.val and is_same_list(p1.next, p2.next)

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def tree_node(values: list):
    if not values:
        return None
    root = TreeNode(values[0])
    i = 1
    queue = deque()
    queue.append(root)
    while queue:
        node = queue.popleft()
        if i < len(values) and values[i] is not None:
            node.left = TreeNode(values[i])
            queue.append(node.left)
        i += 1
        if i < len(values) and values[i] is not None:
            node.right = TreeNode(values[i])
            queue.append(node.right)
        i += 1
    return root

def is_same_tree(p, q):
    if not p and not q:
        return True
    elif not p or not q:
        return False
    elif p.val != q.val:
        return False
    else:
        return is_same_tree(p.left, q.left) and is_same_tree(p.right, q.right)

class Solution:
    def answerString(self, word: str, numFriends: int) -> str:
        if numFriends == 1:
            return word
        n = len(word)
        ans = ""
        for i in range(n):
            k = min(n - i, n - numFriends + 1)
            ans = max(ans, word[i : i + k])
        return ans

class Solution:
    def answerString(self, word: str, numFriends: int) -> str:


You are an expert Python programmer. Write correct, efficient Python 3 code that solves the problem and passes all tests. Output the code in the following format: ```python
<code>
```

PROBLEM:
You are given an array nums consisting of positive integers.
A special subsequence is defined as a subsequence of length 4, represented by indices (p, q, r, s), where p < q < r < s. This subsequence must satisfy the following conditions:

nums[p] * nums[r] == nums[q] * nums[s]
There must be at least one element between each pair of indices. In other words, q - p > 1, r - q > 1 and s - r > 1.

Return the number of different special subsequences in nums.

Example 1:

Input: nums = [1,2,3,4,3,6,1]
Output: 1
Explanation:
There is one special subsequence in nums.

(p, q, r, s) = (0, 2, 4, 6):

This corresponds to elements (1, 3, 3, 1).
nums[p] * nums[r] = nums[0] * nums[4] = 1 * 3 = 3
nums[q] * nums[s] = nums[2] * nums[6] = 3 * 1 = 3

Example 2:

Input: nums = [3,4,3,4,3,4,3,4]
Output: 3
Explanation:
There are three special subsequences in nums.

(p, q, r, s) = (0, 2, 4, 6):

This corresponds to elements (3, 3, 3, 3).
nums[p] * nums[r] = nums[0] * nums[4] = 3 * 3 = 9
nums[q] * nums[s] = nums[2] * nums[6] = 3 * 3 = 9

(p, q, r, s) = (1, 3, 5, 7):

This corresponds to elements (4, 4, 4, 4).
nums[p] * nums[r] = nums[1] * nums[5] = 4 * 4 = 16
nums[q] * nums[s] = nums[3] * nums[7] = 4 * 4 = 16

(p, q, r, s) = (0, 2, 5, 7):

This corresponds to elements (3, 3, 4, 4).
nums[p] * nums[r] = nums[0] * nums[5] = 3 * 4 = 12
nums[q] * nums[s] = nums[2] * nums[7] = 3 * 4 = 12

Constraints:

7 <= nums.length <= 1000
1 <= nums[i] <= 1000

Your solution to the problem should be a method of the class Solution called numberOfSubsequences and should pass all tests. Use the following starter code:
```python
class Solution:
    def numberOfSubsequences(self, nums: List[int]) -> int:
        # Add your code here
```

SOLUTION:

Question:
PROBLEM:
You are given an array nums consisting of positive integers.
A special subsequence is defined as a subsequence of length 4, represented by indices (p, q, r, s), where p < q < r < s. This subsequence must satisfy the following conditions:

nums[p] * nums[r] == nums[q] * nums[s]
There must be at least one element between each pair of indices. In other words, q - p > 1, r - q > 1 and s - r > 1.

Return the number of different special subsequences in nums.

Example 1:

Input: nums = [1,2,3,4,3,6,1]
Output: 1
Explanation:
There is one special subsequence in nums.

(p, q, r, s) = (0, 2, 4, 6):

This corresponds to elements (1, 3, 3, 1).
nums[p] * nums[r] = nums[0] * nums[4] = 1 * 3 = 3
nums[q] * nums[s] = nums[2] * nums[6] = 3 * 1 = 3

Example 2:

Input: nums = [3,4,3,4,3,4,3,4]
Output: 3
Explanation:
There are three special subsequences in nums.

(p, q, r, s) = (0, 2, 4, 6):

This corresponds to elements (3, 3, 3, 3).
nums[p] * nums[r] = nums[0] * nums[4] = 3 * 3 = 9
nums[q] * nums[s] = nums[2] * nums[6] = 3 * 3 = 9

(p, q, r, s) = (1, 3, 5, 7):

This corresponds to elements (4, 4, 4, 4).
nums[p] * nums[r] = nums[1] * nums[5] = 4 * 4 = 16
nums[q] * nums[s] = nums[3] * nums[7] = 4 * 4 = 16

(p, q, r, s) = (0, 2, 5, 7):

This corresponds to elements (3, 3, 4, 4).
nums[p] * nums[r] = nums[0] * nums[5] = 3 * 4 = 12
nums[q] * nums[s] = nums[2] * nums[7] = 3 * 4 = 12

Constraints:

7 <= nums.length <= 1000
1 <= nums[i] <= 1000

Your solution to the problem should be a method of the class Solution called numberOfSubsequences and should pass all tests. Use the following starter code:
```python
class Solution:
    def numberOfSubsequences(self, nums: List[int]) -> int:
        # Add your code here
```

SOLUTION:

assert Solution().numberOfSubsequences(nums = [1,2,3,4,3,6,1]) == 1

assert Solution().numberOfSubsequences(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1]) == 210

assert Solution().numberOfSubsequences(nums = [1,3,5,7,9,11,13,15,17,19]) == 0

assert Solution().numberOfSubsequences(nums = [3,4,3,4,3,4,3,4]) == 3

assert Solution().numberOfSubsequences(nums = [1,1,1,1,1,1,1,1]) == 5

assert Solution().numberOfSubsequences(nums = [2,3,5,7,11,13,17,19]) == 0

assert Solution().numberOfSubsequences(nums = [5,5,5,5,5,5,5,5,5,5]) == 35

assert Solution().numberOfSubsequences(nums = [2,3,2,3,2,3,2,3,2,3]) == 18

assert Solution().numberOfSubsequences(nums = [1, 2, 3, 6, 2, 4, 3, 12, 6, 3, 2, 4]) == 10

Answer: assert Solution().numberOfSubsequences(nums = [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]) == 17550

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,10,20,30,40,10,20,30,40,10]) == 23

Answer: assert Solution().numberOfSubsequences(nums = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [10,20,10,20,10,20,10,20,10,20,10,20,10,20,10,20]) == 315

Answer: assert Solution().numberOfSubsequences(nums = [5,10,15,20,25,30,35,40,45,50,55,60]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [9,3,9,3,9,3,9,3,9,3,9,3,9,3,9,3,9,3]) == 588

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,3,6,1]) == 1

Answer: assert Solution().numberOfSubsequences(nums = [6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6]) == 2380

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,50,60,70,80,90,100,110,120,130,140,150,160,170,180,190,200,210,220,230,240,250]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7]) == 715

Answer: assert Solution().numberOfSubsequences(nums = [7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7]) == 20475

Answer: assert Solution().numberOfSubsequences(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1]) == 210

Answer: assert Solution().numberOfSubsequences(nums = [10,20,10,20,10,20,10,20,10,20,10,20,10,20,10,20,10,20,10,20]) == 1008

Answer: assert Solution().numberOfSubsequences(nums = [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 23751

Answer: assert Solution().numberOfSubsequences(nums = [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]) == 2380

Answer: assert Solution().numberOfSubsequences(nums = [1,3,5,7,9,11,13,15,17,19]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [3,4,3,4,3,4,3,4]) == 3

Answer: assert Solution().numberOfSubsequences(nums = [2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]) == 7315

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,50,60,70,80,90,100,110,120,130,140,150,160,170,180,190,200,210,220,230,240,250,260,270,280,290,300]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1,1,1,1,1,1,1,1]) == 5

Answer: assert Solution().numberOfSubsequences(nums = [2,3,5,7,11,13,17,19]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [100,200,100,200,100,200,100,200,100,200,100,200,100,200,100,200]) == 315

Answer: assert Solution().numberOfSubsequences(nums = [5,5,5,5,5,5,5,5,5,5]) == 35

Answer: assert Solution().numberOfSubsequences(nums = [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [2,3,2,3,2,3,2,3,2,3]) == 18

Answer: assert Solution().numberOfSubsequences(nums = [5,15,5,15,5,15,5,15,5,15,5,15,5,15,5,15,5,15]) == 588

Answer: assert Solution().numberOfSubsequences(nums = [7,21,7,21,7,21,7,21,7,21,7,21,7,21,7,21,7,21]) == 588

Answer: assert Solution().numberOfSubsequences(nums = [100,200,300,400,500,600,700,800,900,1000,100,200,300,400]) == 9

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,1,2,3,4,5,6,7,8,9,10]) == 114

Answer: assert Solution().numberOfSubsequences(nums = [2,6,18,54,162,486,1458,4374,13122,39366,118098,354294,1062882,3188646,9565938,28697814,86093442,258280326,774840978,2324522934]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [7,49,343,2401,16807,117649,7,49,343,2401,16807,117649,7,49,343,2401,16807,117649]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [8,4,2,16,8,4,2,32,16,8,4,2,64,32,16,8,4,2]) == 146

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,50,60,70,80,90,100,5,10,15,20,25,30,35,40,45,50]) == 27

Answer: assert Solution().numberOfSubsequences(nums = [2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 32, 36, 40, 45, 48, 50, 60, 64, 72, 75, 80, 90, 96, 100, 120, 125, 128, 144, 150, 160, 180, 192, 200, 225, 240, 250, 256, 288, 300, 320, 360, 375, 384, 400, 450, 480, 500, 512, 540, 576, 600, 625, 640, 720, 750, 768, 800, 900, 960, 1000, 1024, 1080, 1125, 1152, 1200, 1250, 1280, 1440, 1500, 1536, 1600, 1800, 1920, 2000, 2048, 2160, 2250, 2304, 2400, 2500, 2560, 2880, 3000, 3072, 3200, 3600, 3750, 3840, 4000, 4096, 4320, 4500, 4608, 4800, 5000, 5120, 5400, 5625, 5760, 6000, 6250, 6400, 7200, 7500, 7680, 8000, 8192, 8640, 9000, 9375, 9600, 10000, 10240, 10800, 11250, 11520, 12000, 12500, 12800, 14400, 15000, 15360, 16000, 16200, 16875, 17280, 18000, 18750, 19200, 20000, 20480, 21600, 22500, 23040, 24000, 25000, 25600, 27000, 28125, 28800, 30000, 30720, 32000, 32400, 33750, 34560, 36000, 37500, 38400, 40000, 40960, 43200, 45000, 46080, 48000, 50000, 51200, 54000, 56250, 57600, 60000, 62500, 64000, 72000, 75000, 76800, 80000, 81000, 86400, 90000, 93750, 96000, 100000]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [2,3,6,9,18,27,54,81,162,243,2,3,6,9,18,27,54,81,162,243]) == 96

Answer: assert Solution().numberOfSubsequences(nums = [6,6,6,12,12,12,18,18,18,24,24,24,30,30,30]) == 6

Answer: assert Solution().numberOfSubsequences(nums = [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,50,60,70,80,90,100,10,20,30,40,50,60,70,80,90,100]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,5,6,7,8,9,10,2,4,6,8,10,12,14,16,18,20]) == 26

Answer: assert Solution().numberOfSubsequences(nums = [5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,5,10,15,20,25,30,35,40,45,50]) == 114

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,50,60,70,80,90,100,110,120,130,140,150,160,170,180,190,200,10,20,30,40,50]) == 42

Answer: assert Solution().numberOfSubsequences(nums = [7,7,14,14,28,28,56,56,112,112,7,14,28,56]) == 56

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,2,3,4,2,3,4,2,3,4,2,3,4,2,3,4]) == 245

Answer: assert Solution().numberOfSubsequences(nums = [7,7,14,14,7,14,7,14,7,14,7,14,7,14,7,14,7,14,7,14]) == 945

Answer: assert Solution().numberOfSubsequences(nums = [8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [2,4,8,16,32,64,128,256,512,1024,2,4,8,16]) == 16

Answer: assert Solution().numberOfSubsequences(nums = [3, 3, 9, 9, 3, 9, 9, 3, 9, 3, 9, 3, 9, 3, 9, 3, 9, 3, 9, 3]) == 945

Answer: assert Solution().numberOfSubsequences(nums = [11,22,33,44,55,66,77,88,99,110,121,132,143,154,165,176,187,198,209,220,11,22,33]) == 17

Answer: assert Solution().numberOfSubsequences(nums = [1,1,1,2,2,2,4,4,4,8,8,8,16,16,16,32,32,32]) == 10

Answer: assert Solution().numberOfSubsequences(nums = [3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [2,4,8,16,32,64,128,256,512,1024,512,256,128,64,32,16,8,4,2]) == 252

Answer: assert Solution().numberOfSubsequences(nums = [10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55]) == 20

Answer: assert Solution().numberOfSubsequences(nums = [2,3,4,6,8,12,16,24,32,48,64,96,128,192,256,384,512,768,1024,1536]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 15, 30, 45, 60, 75, 90, 105, 120, 135, 150]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [10, 20, 30, 20, 10, 30, 20, 10, 30, 20, 10, 30, 20, 10, 30]) == 84

Answer: assert Solution().numberOfSubsequences(nums = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [3,6,9,12,15,18,21,24,27,30,3,6,9,12,15,18,21,24,27,30]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [5,25,5,50,5,75,5,100,5,125,5,150,5,175,5,200,5,225,5,250]) == 210

Answer: assert Solution().numberOfSubsequences(nums = [3,9,27,81,243,729,3,9,27,81,243,729,3,9,27,81,243,729]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [1, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1]) == 980

Answer: assert Solution().numberOfSubsequences(nums = [2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2]) == 252

Answer: assert Solution().numberOfSubsequences(nums = [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [2,4,8,16,32,64,128,256,512,1024,2,4,8,16,32,64]) == 44

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 256, 384, 512, 768, 1024]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [10, 20, 10, 30, 40, 30, 50, 60, 50, 70, 80, 70, 90, 100, 90]) == 6

Answer: assert Solution().numberOfSubsequences(nums = [1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,4,4,4,4,4,4,4,4,4,4]) == 3865

Answer: assert Solution().numberOfSubsequences(nums = [10,9,8,7,6,5,4,3,2,1,10,9,8,7,6,5,4,3,2,1]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]) == 2380

Answer: assert Solution().numberOfSubsequences(nums = [4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [5,10,15,20,25,30,35,40,45,50,10,20,30,40,50,60,70,80,90,100]) == 26

Answer: assert Solution().numberOfSubsequences(nums = [9,18,27,36,45,54,63,72,81,90,99,108,117,126,135,144,153,162,171,180]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1,2,4,8,16,32,64,128,256,512,1,2,4,8,16,32,64,128,256,512]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,50,60,70,80,90,100,20,40,60,80,100,120,140,160,180,200]) == 26

Answer: assert Solution().numberOfSubsequences(nums = [7,14,21,28,35,42,49,56,63,70,77,84,91,98,105,112,119,126,133,140,7,14,21,28,35,42,49,56,63,70]) == 114

Answer: assert Solution().numberOfSubsequences(nums = [11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 11, 22, 33, 44, 55, 66, 77, 88, 99, 110]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,100,5,10,15,20,25]) == 42

Answer: assert Solution().numberOfSubsequences(nums = [1, 1, 2, 2, 3, 3, 6, 6, 9, 9, 18, 18, 27, 27, 54, 54, 81, 81, 162, 162, 243, 243, 486, 486, 729, 729, 1458, 1458, 2187, 2187, 4374, 4374, 6561, 6561, 13122, 13122, 19683, 19683, 39366, 39366, 59049, 59049]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [13,26,39,52,65,78,91,104,117,130,143,156,169,182,195,208,221,234,247,260,13,26]) == 8

Answer: assert Solution().numberOfSubsequences(nums = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39,1,3,5,7,9,11,13,15,17,19]) == 38

Answer: assert Solution().numberOfSubsequences(nums = [1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10,1,2,3,4,5,6,7,8,9,10]) == 298

Answer: assert Solution().numberOfSubsequences(nums = [2,4,8,16,32,64,128,256,512,1024,2,4,8,16,32,64,128,256,512,1024]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [5,10,15,20,25,30,35,40,45,50,55,60,65,70,75]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [15, 20, 25, 30, 35, 40, 45, 50, 15, 20, 25, 30, 35, 40, 45, 50, 15, 20, 25, 30]) == 65

Answer: assert Solution().numberOfSubsequences(nums = [1, 1, 2, 2, 3, 3, 4, 4, 6, 6, 8, 8, 12, 12, 16, 16, 24, 24, 32, 32, 48, 48, 64, 64, 96, 96, 128, 128, 192, 192]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48,51,54,57,60,3,6,9,12,15]) == 42

Answer: assert Solution().numberOfSubsequences(nums = [7, 14, 7, 21, 14, 28, 7, 14, 21, 28, 7, 14, 21, 28, 7, 14, 21, 28]) == 151

Answer: assert Solution().numberOfSubsequences(nums = [1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3]) == 3865

Answer: assert Solution().numberOfSubsequences(nums = [5,15,25,35,45,55,65,75,85,95,1,3,5,7,9,11,13,15,17,19]) == 8

Answer: assert Solution().numberOfSubsequences(nums = [1, 2, 3, 6, 2, 4, 3, 12, 6, 3, 2, 4]) == 10

Answer: assert Solution().numberOfSubsequences(nums = [1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [7,14,21,28,35,42,49,56,63,70,77,84,91,98,105,112,119,126,133,140,7,14,21,28]) == 31

Answer: assert Solution().numberOfSubsequences(nums = [1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4, 1, 2, 3, 4]) == 256

Answer: assert Solution().numberOfSubsequences(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [1,4,9,16,25,36,49,64,81,100,1,4,9,16,25,36,49,64,81,100]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [2, 3, 6, 9, 18, 27, 54, 81, 162, 243, 486, 729, 1458, 2187, 4374, 6561, 13122, 19683, 39366, 59049]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [10,20,30,40,50,60,70,80,90,100,110,120,130,140,150,160,170,180,190,200]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [17,34,51,68,85,102,119,136,153,170,187,204,221,238,255,272,289,306,323,340,17,34]) == 8

Answer: assert Solution().numberOfSubsequences(nums = [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683]) == 112

Answer: assert Solution().numberOfSubsequences(nums = [1, 3, 6, 18, 2, 6, 18, 3, 6, 18, 1, 3, 6, 18, 2, 6, 18, 3, 6, 18]) == 227

Answer: assert Solution().numberOfSubsequences(nums = [10,10,20,20,30,30,40,40,50,50,10,20,30,40,50,60,70,80,90,100]) == 30

Answer: assert Solution().numberOfSubsequences(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5]) == 78

Answer: assert Solution().numberOfSubsequences(nums = [9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90]) == 28

Answer: assert Solution().numberOfSubsequences(nums = [500,250,125,625,312,156,78,39,19,9,4,2,1,2,4,8,16,32]) == 25

Answer: assert Solution().numberOfSubsequences(nums = [7, 7, 14, 14, 21, 21, 28, 28, 35, 35, 42, 42, 49, 49, 56, 56, 63, 63, 70, 70, 77, 77, 84, 84, 91, 91, 98, 98]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 7, 14, 21, 28, 35, 42]) == 29

Answer: assert Solution().numberOfSubsequences(nums = [3,6,9,12,15,18,21,24,27,30,33,36,39,42,45]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 177147, 531441, 1594323, 4782969, 14348907, 43046721, 129140163, 387420489, 1162261467, 3486784401, 10460353203, 31381059609, 94143178827, 282429536481, 847288609443, 2541865828329, 7625597484987, 22876792454961, 68630377364883, 205891132094649]) == 0

Answer: assert Solution().numberOfSubsequences(nums = [2,4,8,16,32,64,128,256,512,1024,2048,4096,8192,16384,32768,65536,131072,262144,524288,1048576,2]) == 0

import random
import functools
import collections
import string
import math
import datetime

from typing import *
from functools import *
from collections import *
from itertools import *
from heapq import *
from bisect import *
from string import *
from operator import *
from math import *

inf = float('inf')

class ListNode:
    def __init__(self, val=0, next=None):
        self.val = val
        self.next = next

def list_node(values: list):
    if not values:
        return None
    head = ListNode(values[0])
    p = head
    for val in values[1:]:
        node = ListNode(val)
        p.next = node
        p = node
    return head

def is_same_list(p1, p2):
    if p1 is None and p2 is None:
        return True
    if not p1 or not p2:
        return False
    return p1.val == p2.val and is_same_list(p1.next, p2.next)

class TreeNode:
    def __init__(self, val=0, left=None, right=None):
        self.val = val
        self.left = left
        self.right = right

def tree_node(values: list):
    if not values:
        return None
    root = TreeNode(values[0])
    i = 1
    queue = deque()
    queue.append(root)
    while queue:
        node = queue.popleft()
        if i < len(values) and values[i] is not None:
            node.left = TreeNode(values[i])
            queue.append(node.left)
        i += 1
        if i < len(values) and values[i] is not None:
            node.right = TreeNode(values[i])
            queue.append(node.right)
        i += 1
    return root

def is_same_tree(p, q):
    if not p and not q:
        return True
    elif not p or not q:
        return False
    elif p.val != q.val:
        return False
    else:
        return is_same_tree(p.left, q.left) and is_same_tree(p.right, q.right)

class Solution:
    def numberOfSubsequences(self, nums: List[int]) -> int:
        n = len(nums)
        cnt = defaultdict(int)
        for r in range(4, n - 2):
            c = nums[r]
            for s in range(r + 2, n):
                d = nums[s]
                g = gcd(c, d)
                cnt[(d // g, c // g)] += 1
        ans = 0
        for q in range(2, n - 4):
            b = nums[q]
            for p in range(q - 1):
                a = nums[p]
                g = gcd(a, b)
                ans += cnt[(a // g, b // g)]
            c = nums[q + 2]
            for s in range(q + 4, n):
                d = nums[s]
                g = gcd(c, d)
                cnt[(d // g, c // g)] -= 1
        return ans