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:
Given a positive integer s, let A be a 3D array of dimensions n × n × n, where each element A[i][j][k] is defined as:

A[i][j][k] = i * (j OR k), where 0 <= i, j, k < n.

Return the maximum possible value of n such that the sum of all elements in array A does not exceed s.

Example 1:

Input: s = 10
Output: 2
Explanation:

Elements of the array A for n = 2:

A[0][0][0] = 0 * (0 OR 0) = 0
A[0][0][1] = 0 * (0 OR 1) = 0
A[0][1][0] = 0 * (1 OR 0) = 0
A[0][1][1] = 0 * (1 OR 1) = 0
A[1][0][0] = 1 * (0 OR 0) = 0
A[1][0][1] = 1 * (0 OR 1) = 1
A[1][1][0] = 1 * (1 OR 0) = 1
A[1][1][1] = 1 * (1 OR 1) = 1

The total sum of the elements in array A is 3, which does not exceed 10, so the maximum possible value of n is 2.

Example 2:

Input: s = 0
Output: 1
Explanation:

Elements of the array A for n = 1:

A[0][0][0] = 0 * (0 OR 0) = 0

The total sum of the elements in array A is 0, which does not exceed 0, so the maximum possible value of n is 1.

Constraints:

0 <= s <= 1015

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

SOLUTION:

Question:
PROBLEM:
Given a positive integer s, let A be a 3D array of dimensions n × n × n, where each element A[i][j][k] is defined as:

A[i][j][k] = i * (j OR k), where 0 <= i, j, k < n.

Return the maximum possible value of n such that the sum of all elements in array A does not exceed s.

Example 1:

Input: s = 10
Output: 2
Explanation:

Elements of the array A for n = 2:

A[0][0][0] = 0 * (0 OR 0) = 0
A[0][0][1] = 0 * (0 OR 1) = 0
A[0][1][0] = 0 * (1 OR 0) = 0
A[0][1][1] = 0 * (1 OR 1) = 0
A[1][0][0] = 1 * (0 OR 0) = 0
A[1][0][1] = 1 * (0 OR 1) = 1
A[1][1][0] = 1 * (1 OR 0) = 1
A[1][1][1] = 1 * (1 OR 1) = 1

The total sum of the elements in array A is 3, which does not exceed 10, so the maximum possible value of n is 2.

Example 2:

Input: s = 0
Output: 1
Explanation:

Elements of the array A for n = 1:

A[0][0][0] = 0 * (0 OR 0) = 0

The total sum of the elements in array A is 0, which does not exceed 0, so the maximum possible value of n is 1.

Constraints:

0 <= s <= 1015

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

SOLUTION:

assert Solution().maxSizedArray(s = 100000000000000) == 756

assert Solution().maxSizedArray(s = 10) == 2

assert Solution().maxSizedArray(s = 27) == 2

assert Solution().maxSizedArray(s = 1000) == 5

assert Solution().maxSizedArray(s = 1000000000000000) == 1196

assert Solution().maxSizedArray(s = 123456789) == 50

assert Solution().maxSizedArray(s = 50) == 3

assert Solution().maxSizedArray(s = 1000000000000) == 300

assert Solution().maxSizedArray(s = 1) == 1

assert Solution().maxSizedArray(s = 64) == 3

assert Solution().maxSizedArray(s = 0) == 1

assert Solution().maxSizedArray(s = 100) == 3

assert Solution().maxSizedArray(s = 125) == 3

assert Solution().maxSizedArray(s = 68719476735) == 176

assert Solution().maxSizedArray(s = 200000000000000) == 874

assert Solution().maxSizedArray(s = 1000000000) == 75

assert Solution().maxSizedArray(s = 500) == 4

assert Solution().maxSizedArray(s = 9876543210) == 121

assert Solution().maxSizedArray(s = 4096) == 6

assert Solution().maxSizedArray(s = 2) == 1

assert Solution().maxSizedArray(s = 134217727) == 51

assert Solution().maxSizedArray(s = 1024) == 5

assert Solution().maxSizedArray(s = 999999999999999) == 1196

assert Solution().maxSizedArray(s = 268435455) == 59

assert Solution().maxSizedArray(s = 1000000000000001) == 1196

assert Solution().maxSizedArray(s = 4095) == 6

assert Solution().maxSizedArray(s = 511) == 4

assert Solution().maxSizedArray(s = 67890123456789) == 698

assert Solution().maxSizedArray(s = 32) == 2

assert Solution().maxSizedArray(s = 123456789012345) == 790

assert Solution().maxSizedArray(s = 18) == 2

assert Solution().maxSizedArray(s = 15) == 2

assert Solution().maxSizedArray(s = 128) == 3

assert Solution().maxSizedArray(s = 2048) == 5

assert Solution().maxSizedArray(s = 29) == 2

assert Solution().maxSizedArray(s = 8) == 2

assert Solution().maxSizedArray(s = 300000000000000) == 952

assert Solution().maxSizedArray(s = 256) == 4

assert Solution().maxSizedArray(s = 127) == 3

assert Solution().maxSizedArray(s = 999999999999) == 300

assert Solution().maxSizedArray(s = 100000000000) == 190

assert Solution().maxSizedArray(s = 10000) == 8

assert Solution().maxSizedArray(s = 42) == 2

assert Solution().maxSizedArray(s = 999999999999998) == 1196

assert Solution().maxSizedArray(s = 7) == 2

assert Solution().maxSizedArray(s = 16) == 2

assert Solution().maxSizedArray(s = 255) == 4

assert Solution().maxSizedArray(s = 4294967295) == 101

assert Solution().maxSizedArray(s = 1025) == 5

assert Solution().maxSizedArray(s = 5000000000) == 105

assert Solution().maxSizedArray(s = 67108863) == 44

assert Solution().maxSizedArray(s = 8191) == 7

assert Solution().maxSizedArray(s = 1048576) == 19

assert Solution().maxSizedArray(s = 100000) == 12

assert Solution().maxSizedArray(s = 100000000) == 48

assert Solution().maxSizedArray(s = 1099511627775) == 306

assert Solution().maxSizedArray(s = 512) == 4

assert Solution().maxSizedArray(s = 31) == 2

assert Solution().maxSizedArray(s = 1023) == 5

assert Solution().maxSizedArray(s = 28) == 2

assert Solution().maxSizedArray(s = 16383) == 8

assert Solution().maxSizedArray(s = 500000000000000) == 1054

assert Solution().maxSizedArray(s = 1000000) == 19

Answer: assert Solution().maxSizedArray(s = 100000000000000) == 756

Answer: assert Solution().maxSizedArray(s = 10) == 2

Answer: assert Solution().maxSizedArray(s = 27) == 2

Answer: assert Solution().maxSizedArray(s = 1000) == 5

Answer: assert Solution().maxSizedArray(s = 1000000000000000) == 1196

Answer: assert Solution().maxSizedArray(s = 123456789) == 50

Answer: assert Solution().maxSizedArray(s = 50) == 3

Answer: assert Solution().maxSizedArray(s = 1000000000000) == 300

Answer: assert Solution().maxSizedArray(s = 1) == 1

Answer: assert Solution().maxSizedArray(s = 64) == 3

Answer: assert Solution().maxSizedArray(s = 0) == 1

Answer: assert Solution().maxSizedArray(s = 100) == 3

Answer: assert Solution().maxSizedArray(s = 125) == 3

Answer: assert Solution().maxSizedArray(s = 68719476735) == 176

Answer: assert Solution().maxSizedArray(s = 200000000000000) == 874

Answer: assert Solution().maxSizedArray(s = 1000000000) == 75

Answer: assert Solution().maxSizedArray(s = 500) == 4

Answer: assert Solution().maxSizedArray(s = 9876543210) == 121

Answer: assert Solution().maxSizedArray(s = 4096) == 6

Answer: assert Solution().maxSizedArray(s = 9223372036854775807) == 1330

Answer: assert Solution().maxSizedArray(s = 2) == 1

Answer: assert Solution().maxSizedArray(s = 134217727) == 51

Answer: assert Solution().maxSizedArray(s = 1024) == 5

Answer: assert Solution().maxSizedArray(s = 999999999999999) == 1196

Answer: assert Solution().maxSizedArray(s = 268435455) == 59

Answer: assert Solution().maxSizedArray(s = 1000000000000001) == 1196

Answer: assert Solution().maxSizedArray(s = 4095) == 6

Answer: assert Solution().maxSizedArray(s = 511) == 4

Answer: assert Solution().maxSizedArray(s = 987654321987654321) == 1330

Answer: assert Solution().maxSizedArray(s = 67890123456789) == 698

Answer: assert Solution().maxSizedArray(s = 32) == 2

Answer: assert Solution().maxSizedArray(s = 1000000000000000000) == 1330

Answer: assert Solution().maxSizedArray(s = 123456789012345) == 790

Answer: assert Solution().maxSizedArray(s = 18) == 2

Answer: assert Solution().maxSizedArray(s = 15) == 2

Answer: assert Solution().maxSizedArray(s = 128) == 3

Answer: assert Solution().maxSizedArray(s = 2048) == 5

Answer: assert Solution().maxSizedArray(s = 29) == 2

Answer: assert Solution().maxSizedArray(s = 1234567890123456789) == 1330

Answer: assert Solution().maxSizedArray(s = 8) == 2

Answer: assert Solution().maxSizedArray(s = 300000000000000) == 952

Answer: assert Solution().maxSizedArray(s = 256) == 4

Answer: assert Solution().maxSizedArray(s = 127) == 3

Answer: assert Solution().maxSizedArray(s = 999999999999) == 300

Answer: assert Solution().maxSizedArray(s = 100000000000) == 190

Answer: assert Solution().maxSizedArray(s = 10000) == 8

Answer: assert Solution().maxSizedArray(s = 42) == 2

Answer: assert Solution().maxSizedArray(s = 999999999999998) == 1196

Answer: assert Solution().maxSizedArray(s = 7) == 2

Answer: assert Solution().maxSizedArray(s = 16) == 2

Answer: assert Solution().maxSizedArray(s = 255) == 4

Answer: assert Solution().maxSizedArray(s = 4294967295) == 101

Answer: assert Solution().maxSizedArray(s = 1025) == 5

Answer: assert Solution().maxSizedArray(s = 5000000000) == 105

Answer: assert Solution().maxSizedArray(s = 67108863) == 44

Answer: assert Solution().maxSizedArray(s = 8191) == 7

Answer: assert Solution().maxSizedArray(s = 18446744073709551615) == 1330

Answer: assert Solution().maxSizedArray(s = 1048576) == 19

Answer: assert Solution().maxSizedArray(s = 100000) == 12

Answer: assert Solution().maxSizedArray(s = 100000000) == 48

Answer: assert Solution().maxSizedArray(s = 1099511627775) == 306

Answer: assert Solution().maxSizedArray(s = 512) == 4

Answer: assert Solution().maxSizedArray(s = 31) == 2

Answer: assert Solution().maxSizedArray(s = 1023) == 5

Answer: assert Solution().maxSizedArray(s = 10000000000000000) == 1330

Answer: assert Solution().maxSizedArray(s = 28) == 2

Answer: assert Solution().maxSizedArray(s = 16383) == 8

Answer: assert Solution().maxSizedArray(s = 500000000000000) == 1054

Answer: assert Solution().maxSizedArray(s = 1000000) == 19

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)

mx = 1330
f = [0] * mx
for i in range(1, mx):
    f[i] = f[i - 1] + i
    for j in range(i):
        f[i] += 2 * (i | j)

class Solution:
    def maxSizedArray(self, s: int) -> int:
        l, r = 1, mx
        while l < r:
            m = (l + r + 1) >> 1
            if f[m - 1] * (m - 1) * m // 2 <= s:
                l = m
            else:
                r = m - 1
        return l


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 integer array nums and two integers k and numOperations.
You must perform an operation numOperations times on nums, where in each operation you:

Select an index i that was not selected in any previous operations.
Add an integer in the range [-k, k] to nums[i].

Return the maximum possible frequency of any element in nums after performing the operations.

Example 1:

Input: nums = [1,4,5], k = 1, numOperations = 2
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1]. nums becomes [1, 4, 5].
Adding -1 to nums[2]. nums becomes [1, 4, 4].

Example 2:

Input: nums = [5,11,20,20], k = 5, numOperations = 1
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1].

Constraints:

1 <= nums.length <= 105
1 <= nums[i] <= 105
0 <= k <= 105
0 <= numOperations <= nums.length

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

SOLUTION:

Question:
PROBLEM:
You are given an integer array nums and two integers k and numOperations.
You must perform an operation numOperations times on nums, where in each operation you:

Select an index i that was not selected in any previous operations.
Add an integer in the range [-k, k] to nums[i].

Return the maximum possible frequency of any element in nums after performing the operations.

Example 1:

Input: nums = [1,4,5], k = 1, numOperations = 2
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1]. nums becomes [1, 4, 5].
Adding -1 to nums[2]. nums becomes [1, 4, 4].

Example 2:

Input: nums = [5,11,20,20], k = 5, numOperations = 1
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1].

Constraints:

1 <= nums.length <= 105
1 <= nums[i] <= 105
0 <= k <= 105
0 <= numOperations <= nums.length

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

SOLUTION:

assert Solution().maxFrequency(nums = [1,100000], k = 100000, numOperations = 1) == 2

assert Solution().maxFrequency(nums = [1,1,1,1,1], k = 10, numOperations = 0) == 5

assert Solution().maxFrequency(nums = [1,100,1000,10000], k = 5000, numOperations = 3) == 3

assert Solution().maxFrequency(nums = [1,4,5], k = 1, numOperations = 2) == 2

assert Solution().maxFrequency(nums = [1,2,2,3,3,3,4], k = 1, numOperations = 4) == 6

assert Solution().maxFrequency(nums = [10,20,30,40,50], k = 15, numOperations = 5) == 4

assert Solution().maxFrequency(nums = [1,1,1,1], k = 10, numOperations = 4) == 4

assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 2, numOperations = 5) == 5

assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 2, numOperations = 3) == 4

assert Solution().maxFrequency(nums = [10,10,10], k = 0, numOperations = 3) == 3

assert Solution().maxFrequency(nums = [10,10,10,10], k = 0, numOperations = 3) == 4

assert Solution().maxFrequency(nums = [10,10,10,10], k = 3, numOperations = 4) == 4

assert Solution().maxFrequency(nums = [5,11,20,20], k = 5, numOperations = 1) == 2

assert Solution().maxFrequency(nums = [1,2,3,4,5,6,7,8,9,10], k = 5, numOperations = 15) == 10

assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 2, numOperations = 15) == 10

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 1, numOperations = 9) == 3

assert Solution().maxFrequency(nums = [5,5,5,5,5,5,5,5,5,5], k = 10, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 5, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 1, numOperations = 20) == 10

assert Solution().maxFrequency(nums = [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3], k = 3, numOperations = 15) == 12

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 1, numOperations = 10) == 2

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 2, numOperations = 10) == 5

assert Solution().maxFrequency(nums = [1,100,200,300,400,500], k = 150, numOperations = 3) == 3

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 1, numOperations = 15) == 2

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 8, numOperations = 9) == 9

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 2, numOperations = 15) == 5

assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 2, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1,1,2,2,3,3,4,4,5,5], k = 3, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 5, numOperations = 5) == 2

assert Solution().maxFrequency(nums = [5, 10, 15, 20, 25], k = 10, numOperations = 5) == 5

assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 15, numOperations = 5) == 4

assert Solution().maxFrequency(nums = [1,3,5,7,9,11,13,15,17,19], k = 4, numOperations = 10) == 5

assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 3, numOperations = 20) == 10

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 3, numOperations = 10) == 7

assert Solution().maxFrequency(nums = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], k = 2, numOperations = 10) == 3

assert Solution().maxFrequency(nums = [3,5,7,9,11], k = 2, numOperations = 5) == 3

assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 2, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 3, numOperations = 15) == 7

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 4, numOperations = 20) == 5

assert Solution().maxFrequency(nums = [10, 10, 20, 20, 30, 30, 40, 40], k = 10, numOperations = 10) == 6

assert Solution().maxFrequency(nums = [1, 3, 6, 9, 12], k = 3, numOperations = 5) == 3

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 5, numOperations = 25) == 6

assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 0, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 10, 100, 1000, 10000], k = 1000, numOperations = 10) == 4

assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 4, 4, 5], k = 1, numOperations = 5) == 7

assert Solution().maxFrequency(nums = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], k = 3, numOperations = 30) == 4

Answer: assert Solution().maxFrequency(nums = [1,100000], k = 100000, numOperations = 1) == 2

Answer: assert Solution().maxFrequency(nums = [1,1,1,1,1], k = 10, numOperations = 0) == 5

Answer: assert Solution().maxFrequency(nums = [1,100,1000,10000], k = 5000, numOperations = 3) == 3

Answer: assert Solution().maxFrequency(nums = [1,4,5], k = 1, numOperations = 2) == 2

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

Answer: assert Solution().maxFrequency(nums = [10,20,30,40,50], k = 15, numOperations = 5) == 4

Answer: assert Solution().maxFrequency(nums = [1,1,1,1], k = 10, numOperations = 4) == 4

Answer: assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 2, numOperations = 5) == 5

Answer: assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 2, numOperations = 3) == 4

Answer: assert Solution().maxFrequency(nums = [10,10,10], k = 0, numOperations = 3) == 3

Answer: assert Solution().maxFrequency(nums = [10,10,10,10], k = 0, numOperations = 3) == 4

Answer: assert Solution().maxFrequency(nums = [10,10,10,10], k = 3, numOperations = 4) == 4

Answer: assert Solution().maxFrequency(nums = [5,11,20,20], k = 5, numOperations = 1) == 2

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 10, numOperations = 50) == 20

Answer: assert Solution().maxFrequency(nums = [1,2,3,4,5,6,7,8,9,10], k = 5, numOperations = 15) == 10

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 2, numOperations = 15) == 10

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500], k = 50, numOperations = 15) == 2

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

Answer: assert Solution().maxFrequency(nums = [50, 50, 50, 50, 50, 50, 50, 50, 50, 50], k = 25, numOperations = 50) == 10

Answer: assert Solution().maxFrequency(nums = [5,5,5,5,5,5,5,5,5,5], k = 10, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10], k = 2, numOperations = 25) == 15

Answer: assert Solution().maxFrequency(nums = [50, 100, 150, 200, 250, 300, 350, 400, 450, 500], k = 50, numOperations = 50) == 3

Answer: assert Solution().maxFrequency(nums = [10, 10, 10, 10, 20, 20, 20, 20, 30, 30, 30, 30], k = 10, numOperations = 15) == 12

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

Answer: assert Solution().maxFrequency(nums = [100,100,100,100,100,100,100], k = 100, numOperations = 7) == 7

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1, numOperations = 100) == 20

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5], k = 2, numOperations = 50) == 20

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1, numOperations = 19) == 20

Answer: assert Solution().maxFrequency(nums = [100000, 100000, 100000, 100000, 100000, 100000, 100000, 100000, 100000, 100000], k = 0, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 1, numOperations = 20) == 10

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 100, numOperations = 20) == 3

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3], k = 3, numOperations = 15) == 12

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 1, numOperations = 10) == 2

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

Answer: assert Solution().maxFrequency(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, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 50], k = 2, numOperations = 50) == 10

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

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

Answer: assert Solution().maxFrequency(nums = [1,100,200,300,400,500], k = 150, numOperations = 3) == 3

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5], k = 1, numOperations = 20) == 15

Answer: assert Solution().maxFrequency(nums = [100000, 99999, 99998, 99997, 99996, 99995, 99994, 99993, 99992, 99991], k = 50000, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1,2,2,3,3,4,4,4,5,5,5,5,6,6,6,7,7,8,8,9], k = 3, numOperations = 20) == 18

Answer: assert Solution().maxFrequency(nums = [1, 10, 100, 1000, 10000, 100000], k = 100000, numOperations = 100) == 6

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10], k = 2, numOperations = 30) == 15

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 1, numOperations = 15) == 2

Answer: assert Solution().maxFrequency(nums = [50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50], k = 10, numOperations = 20) == 20

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 1, numOperations = 100) == 3

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 100, numOperations = 15) == 3

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 8, numOperations = 9) == 9

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 0, numOperations = 0) == 20

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 2, numOperations = 15) == 5

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 2, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1,1,2,2,3,3,4,4,5,5], k = 3, numOperations = 10) == 10

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

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], k = 5, numOperations = 25) == 11

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 2, numOperations = 20) == 10

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5], k = 3, numOperations = 25) == 15

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 5, numOperations = 5) == 2

Answer: assert Solution().maxFrequency(nums = [50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50], k = 20, numOperations = 19) == 20

Answer: assert Solution().maxFrequency(nums = [100000, 90000, 80000, 70000, 60000, 50000, 40000, 30000, 20000, 10000], k = 5000, numOperations = 15) == 2

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5], k = 1, numOperations = 15) == 12

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 0, numOperations = 20) == 20

Answer: assert Solution().maxFrequency(nums = [5, 10, 15, 20, 25], k = 10, numOperations = 5) == 5

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

Answer: assert Solution().maxFrequency(nums = [10, 10, 10, 20, 20, 20, 30, 30, 30, 40], k = 5, numOperations = 15) == 6

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 15, numOperations = 5) == 4

Answer: assert Solution().maxFrequency(nums = [1,3,5,7,9,11,13,15,17,19], k = 4, numOperations = 10) == 5

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5], k = 2, numOperations = 50) == 30

Answer: assert Solution().maxFrequency(nums = [50, 50, 50, 100, 100, 100, 150, 150, 150, 200, 200, 200], k = 50, numOperations = 30) == 9

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 3, numOperations = 20) == 10

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

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200], k = 50, numOperations = 20) == 11

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

Answer: assert Solution().maxFrequency(nums = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], k = 2, numOperations = 10) == 3

Answer: assert Solution().maxFrequency(nums = [3,5,7,9,11], k = 2, numOperations = 5) == 3

Answer: assert Solution().maxFrequency(nums = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 0, numOperations = 19) == 20

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 150, numOperations = 5) == 4

Answer: assert Solution().maxFrequency(nums = [1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000], k = 500, numOperations = 5) == 2

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 7, numOperations = 25) == 15

Answer: assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45], k = 2, numOperations = 11) == 2

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 10, numOperations = 30) == 20

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 2, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7], k = 3, numOperations = 30) == 21

Answer: assert Solution().maxFrequency(nums = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10], k = 5, numOperations = 5) == 10

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 3, numOperations = 15) == 7

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 4, numOperations = 20) == 5

Answer: assert Solution().maxFrequency(nums = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 0, numOperations = 100) == 112

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], k = 20, numOperations = 10) == 5

Answer: assert Solution().maxFrequency(nums = [10, 10, 20, 20, 30, 30, 40, 40], k = 10, numOperations = 10) == 6

Answer: assert Solution().maxFrequency(nums = [1, 3, 6, 9, 12], k = 3, numOperations = 5) == 3

Answer: assert Solution().maxFrequency(nums = [100000, 100000, 100000, 100000, 100000], k = 100000, numOperations = 5) == 5

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 5, numOperations = 25) == 6

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 0, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [10,20,30,40,50,60,70,80,90,100], k = 30, numOperations = 5) == 6

Answer: assert Solution().maxFrequency(nums = [5, 15, 25, 35, 45, 55, 65, 75, 85, 95], k = 20, numOperations = 20) == 5

Answer: assert Solution().maxFrequency(nums = [1, 10, 100, 1000, 10000], k = 1000, numOperations = 10) == 4

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

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 100, numOperations = 50) == 3

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 4, 4, 5], k = 1, numOperations = 5) == 7

Answer: assert Solution().maxFrequency(nums = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], k = 3, numOperations = 30) == 4

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 maxFrequency(self, nums: List[int], k: int, numOperations: int) -> int:
        cnt = defaultdict(int)
        d = defaultdict(int)
        for x in nums:
            cnt[x] += 1
            d[x] += 0
            d[x - k] += 1
            d[x + k + 1] -= 1
        ans = s = 0
        for x, t in sorted(d.items()):
            s += t
            ans = max(ans, min(s, cnt[x] + numOperations))
        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 an integer array nums and two integers k and numOperations.
You must perform an operation numOperations times on nums, where in each operation you:

Select an index i that was not selected in any previous operations.
Add an integer in the range [-k, k] to nums[i].

Return the maximum possible frequency of any element in nums after performing the operations.

Example 1:

Input: nums = [1,4,5], k = 1, numOperations = 2
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1], after which nums becomes [1, 4, 5].
Adding -1 to nums[2], after which nums becomes [1, 4, 4].

Example 2:

Input: nums = [5,11,20,20], k = 5, numOperations = 1
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1].

Constraints:

1 <= nums.length <= 105
1 <= nums[i] <= 109
0 <= k <= 109
0 <= numOperations <= nums.length

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

SOLUTION:

Question:
PROBLEM:
You are given an integer array nums and two integers k and numOperations.
You must perform an operation numOperations times on nums, where in each operation you:

Select an index i that was not selected in any previous operations.
Add an integer in the range [-k, k] to nums[i].

Return the maximum possible frequency of any element in nums after performing the operations.

Example 1:

Input: nums = [1,4,5], k = 1, numOperations = 2
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1], after which nums becomes [1, 4, 5].
Adding -1 to nums[2], after which nums becomes [1, 4, 4].

Example 2:

Input: nums = [5,11,20,20], k = 5, numOperations = 1
Output: 2
Explanation:
We can achieve a maximum frequency of two by:

Adding 0 to nums[1].

Constraints:

1 <= nums.length <= 105
1 <= nums[i] <= 109
0 <= k <= 109
0 <= numOperations <= nums.length

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

SOLUTION:

assert Solution().maxFrequency(nums = [1,4,5], k = 1, numOperations = 2) == 2

assert Solution().maxFrequency(nums = [1,1000000000], k = 1000000000, numOperations = 1) == 2

assert Solution().maxFrequency(nums = [10,10,10,10], k = 0, numOperations = 4) == 4

assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 10, numOperations = 5) == 5

assert Solution().maxFrequency(nums = [1,1000000000], k = 500000000, numOperations = 1) == 1

assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 2, numOperations = 3) == 4

assert Solution().maxFrequency(nums = [10,10,10,10], k = 0, numOperations = 3) == 4

assert Solution().maxFrequency(nums = [5,11,20,20], k = 5, numOperations = 1) == 2

assert Solution().maxFrequency(nums = [1, 2, 2, 2, 3, 3, 4, 4, 5, 5], k = 1, numOperations = 9) == 7

assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1, numOperations = 5) == 10

assert Solution().maxFrequency(nums = [5, 8, 10, 15, 20], k = 5, numOperations = 10) == 4

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9], k = 2, numOperations = 5) == 3

assert Solution().maxFrequency(nums = [3, 9, 12, 15, 20], k = 4, numOperations = 5) == 3

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 4, numOperations = 30) == 9

assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17], k = 4, numOperations = 5) == 3

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6], k = 2, numOperations = 5) == 5

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 5, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 4, numOperations = 15) == 9

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5], k = 10, numOperations = 10) == 5

assert Solution().maxFrequency(nums = [10, 10, 20, 20, 30, 30, 40, 40], k = 10, numOperations = 12) == 6

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 1, numOperations = 20) == 3

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 5, numOperations = 15) == 10

assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 6) == 9

assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17, 21], k = 4, numOperations = 10) == 3

assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 0, numOperations = 5) == 1

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 2, numOperations = 10) == 5

assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 10, numOperations = 5) == 3

assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 2, 2, 2, 2, 2], k = 1, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500], k = 50, numOperations = 7) == 2

assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 10) == 9

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 1, numOperations = 10) == 3

assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17, 21], k = 4, numOperations = 6) == 3

assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 10, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 1, 1, 1, 100], k = 99, numOperations = 5) == 5

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9], k = 3, numOperations = 5) == 4

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 5, numOperations = 20) == 10

assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4], k = 1, numOperations = 10) == 11

assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500], k = 150, numOperations = 7) == 4

assert Solution().maxFrequency(nums = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 2, numOperations = 5) == 10

assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 15) == 9

assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 1, numOperations = 9) == 6

assert Solution().maxFrequency(nums = [1, 2, 3, 3, 3, 3, 4, 5, 6, 7], k = 2, numOperations = 8) == 8

assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4], k = 1, numOperations = 5) == 6

assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 4], k = 1, numOperations = 5) == 7

assert Solution().maxFrequency(nums = [5, 10, 15, 20, 25, 30], k = 7, numOperations = 15) == 3

assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 100, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 3, numOperations = 10) == 7

assert Solution().maxFrequency(nums = [1, 1, 2, 3, 5, 8, 13, 21, 34], k = 10, numOperations = 20) == 8

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 0, numOperations = 0) == 1

assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 15) == 9

assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 10, numOperations = 9) == 10

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5], k = 2, numOperations = 3) == 4

assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 10, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [10, 15, 20, 25, 30], k = 5, numOperations = 4) == 3

assert Solution().maxFrequency(nums = [1, 5, 8, 12, 16], k = 3, numOperations = 4) == 2

assert Solution().maxFrequency(nums = [1, 5, 9, 14, 20], k = 3, numOperations = 5) == 2

assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 3, numOperations = 15) == 7

assert Solution().maxFrequency(nums = [1, 100, 200, 300, 400, 500], k = 50, numOperations = 10) == 2

assert Solution().maxFrequency(nums = [1, 2, 2, 3, 4, 4, 5], k = 3, numOperations = 10) == 7

assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 0, numOperations = 10) == 10

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9], k = 2, numOperations = 3) == 3

assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 1, numOperations = 10) == 6

assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 2, numOperations = 10) == 3

assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60], k = 15, numOperations = 20) == 4

Answer: assert Solution().maxFrequency(nums = [1,4,5], k = 1, numOperations = 2) == 2

Answer: assert Solution().maxFrequency(nums = [1,1000000000], k = 1000000000, numOperations = 1) == 2

Answer: assert Solution().maxFrequency(nums = [10,10,10,10], k = 0, numOperations = 4) == 4

Answer: assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 10, numOperations = 5) == 5

Answer: assert Solution().maxFrequency(nums = [1,1000000000], k = 500000000, numOperations = 1) == 1

Answer: assert Solution().maxFrequency(nums = [1,2,3,4,5], k = 2, numOperations = 3) == 4

Answer: assert Solution().maxFrequency(nums = [10,10,10,10], k = 0, numOperations = 3) == 4

Answer: assert Solution().maxFrequency(nums = [5,11,20,20], k = 5, numOperations = 1) == 2

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150], k = 15, numOperations = 15) == 4

Answer: assert Solution().maxFrequency(nums = [1, 2, 2, 2, 3, 3, 4, 4, 5, 5], k = 1, numOperations = 9) == 7

Answer: assert Solution().maxFrequency(nums = [10, 10, 20, 20, 30, 30, 40, 40, 50, 50], k = 15, numOperations = 20) == 8

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1, numOperations = 5) == 10

Answer: assert Solution().maxFrequency(nums = [5, 8, 10, 15, 20], k = 5, numOperations = 10) == 4

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9], k = 2, numOperations = 5) == 3

Answer: assert Solution().maxFrequency(nums = [500000000, 500000001, 500000002, 500000003, 500000004], k = 2, numOperations = 5) == 5

Answer: assert Solution().maxFrequency(nums = [3, 9, 12, 15, 20], k = 4, numOperations = 5) == 3

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 4, numOperations = 30) == 9

Answer: assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17], k = 4, numOperations = 5) == 3

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6], k = 2, numOperations = 5) == 5

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5], k = 1, numOperations = 15) == 11

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

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 4, numOperations = 15) == 9

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5], k = 10, numOperations = 10) == 5

Answer: assert Solution().maxFrequency(nums = [10, 10, 20, 20, 30, 30, 40, 40], k = 10, numOperations = 12) == 6

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], k = 15, numOperations = 20) == 4

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 1, numOperations = 20) == 3

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 5, numOperations = 15) == 10

Answer: assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 6) == 9

Answer: assert Solution().maxFrequency(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], k = 2, numOperations = 25) == 5

Answer: assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17, 21], k = 4, numOperations = 10) == 3

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 0, numOperations = 5) == 1

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

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50], k = 10, numOperations = 5) == 3

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 100, numOperations = 10) == 3

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

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5], k = 1, numOperations = 40) == 15

Answer: assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57], k = 5, numOperations = 20) == 3

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 2, 2, 2, 2, 2], k = 1, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500], k = 50, numOperations = 7) == 2

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 0, numOperations = 10) == 20

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 50, numOperations = 10) == 2

Answer: assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 10) == 9

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

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], k = 15, numOperations = 15) == 4

Answer: assert Solution().maxFrequency(nums = [1, 5, 9, 13, 17, 21], k = 4, numOperations = 6) == 3

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 10, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 100], k = 99, numOperations = 5) == 5

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9], k = 3, numOperations = 5) == 4

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 5, numOperations = 20) == 10

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 50, numOperations = 15) == 2

Answer: assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4], k = 1, numOperations = 10) == 11

Answer: assert Solution().maxFrequency(nums = [1, 10, 20, 30, 40, 50, 60, 70, 80, 90], k = 5, numOperations = 0) == 1

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 0, numOperations = 0) == 15

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500], k = 150, numOperations = 7) == 4

Answer: assert Solution().maxFrequency(nums = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 2, numOperations = 5) == 10

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

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 1, numOperations = 30) == 21

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5], k = 2, numOperations = 25) == 25

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 15) == 9

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5], k = 1, numOperations = 15) == 12

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 1, numOperations = 9) == 6

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 100, numOperations = 5) == 3

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 3, 3, 3, 4, 5, 6, 7], k = 2, numOperations = 8) == 8

Answer: assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4], k = 1, numOperations = 5) == 6

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 4], k = 1, numOperations = 5) == 7

Answer: assert Solution().maxFrequency(nums = [5, 10, 15, 20, 25, 30], k = 7, numOperations = 15) == 3

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], k = 7, numOperations = 100) == 15

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 100, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 100, numOperations = 25) == 3

Answer: assert Solution().maxFrequency(nums = [1, 10, 20, 30, 40, 50, 60, 70, 80, 90], k = 10, numOperations = 5) == 3

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

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 3, 5, 8, 13, 21, 34], k = 10, numOperations = 20) == 8

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], k = 5, numOperations = 20) == 2

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

Answer: assert Solution().maxFrequency(nums = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4], k = 1, numOperations = 15) == 9

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

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 10, numOperations = 9) == 10

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5], k = 2, numOperations = 3) == 4

Answer: assert Solution().maxFrequency(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 10, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], k = 7, numOperations = 20) == 15

Answer: assert Solution().maxFrequency(nums = [10, 15, 20, 25, 30], k = 5, numOperations = 4) == 3

Answer: assert Solution().maxFrequency(nums = [1, 5, 8, 12, 16], k = 3, numOperations = 4) == 2

Answer: assert Solution().maxFrequency(nums = [1, 5, 9, 14, 20], k = 3, numOperations = 5) == 2

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 3, numOperations = 15) == 7

Answer: assert Solution().maxFrequency(nums = [1, 100, 200, 300, 400, 500], k = 50, numOperations = 10) == 2

Answer: assert Solution().maxFrequency(nums = [1, 2, 2, 3, 4, 4, 5], k = 3, numOperations = 10) == 7

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

Answer: assert Solution().maxFrequency(nums = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50], k = 10, numOperations = 25) == 5

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5], k = 1, numOperations = 15) == 15

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 0, numOperations = 10) == 15

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], k = 5, numOperations = 50) == 2

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

Answer: assert Solution().maxFrequency(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 0, numOperations = 10) == 10

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9], k = 2, numOperations = 3) == 3

Answer: assert Solution().maxFrequency(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], k = 1, numOperations = 10) == 6

Answer: assert Solution().maxFrequency(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 2, numOperations = 10) == 3

Answer: assert Solution().maxFrequency(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], k = 5, numOperations = 30) == 11

Answer: assert Solution().maxFrequency(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 100, numOperations = 50) == 3

Answer: assert Solution().maxFrequency(nums = [10, 20, 30, 40, 50, 60], k = 15, numOperations = 20) == 4

Answer: assert Solution().maxFrequency(nums = [1000000000, 999999999, 999999998, 999999997], k = 500000000, numOperations = 3) == 4

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 maxFrequency(self, nums: List[int], k: int, numOperations: int) -> int:
        cnt = defaultdict(int)
        d = defaultdict(int)
        for x in nums:
            cnt[x] += 1
            d[x] += 0
            d[x - k] += 1
            d[x + k + 1] -= 1
        ans = s = 0
        for x, t in sorted(d.items()):
            s += t
            ans = max(ans, min(s, cnt[x] + numOperations))
        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 num which represents a positive integer, and an integer t.
A number is called zero-free if none of its digits are 0.
Return a string representing the smallest zero-free number greater than or equal to num such that the product of its digits is divisible by t. If no such number exists, return "-1".

Example 1:

Input: num = "1234", t = 256
Output: "1488"
Explanation:
The smallest zero-free number that is greater than 1234 and has the product of its digits divisible by 256 is 1488, with the product of its digits equal to 256.

Example 2:

Input: num = "12355", t = 50
Output: "12355"
Explanation:
12355 is already zero-free and has the product of its digits divisible by 50, with the product of its digits equal to 150.

Example 3:

Input: num = "11111", t = 26
Output: "-1"
Explanation:
No number greater than 11111 has the product of its digits divisible by 26.

Constraints:

2 <= num.length <= 2 * 105
num consists only of digits in the range ['0', '9'].
num does not contain leading zeros.
1 <= t <= 1014

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

SOLUTION:

Question:
PROBLEM:
You are given a string num which represents a positive integer, and an integer t.
A number is called zero-free if none of its digits are 0.
Return a string representing the smallest zero-free number greater than or equal to num such that the product of its digits is divisible by t. If no such number exists, return "-1".

Example 1:

Input: num = "1234", t = 256
Output: "1488"
Explanation:
The smallest zero-free number that is greater than 1234 and has the product of its digits divisible by 256 is 1488, with the product of its digits equal to 256.

Example 2:

Input: num = "12355", t = 50
Output: "12355"
Explanation:
12355 is already zero-free and has the product of its digits divisible by 50, with the product of its digits equal to 150.

Example 3:

Input: num = "11111", t = 26
Output: "-1"
Explanation:
No number greater than 11111 has the product of its digits divisible by 26.

Constraints:

2 <= num.length <= 2 * 105
num consists only of digits in the range ['0', '9'].
num does not contain leading zeros.
1 <= t <= 1014

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

SOLUTION:

assert Solution().smallestNumber(num = "99999", t = 987654) == "-1"

assert Solution().smallestNumber(num = "1000", t = 10) == "1125"

assert Solution().smallestNumber(num = "101010", t = 10) == "111125"

assert Solution().smallestNumber(num = "202020", t = 20) == "211125"

assert Solution().smallestNumber(num = "1999", t = 2000) == "25558"

assert Solution().smallestNumber(num = "1234", t = 256) == "1488"

assert Solution().smallestNumber(num = "1001", t = 10) == "1125"

assert Solution().smallestNumber(num = "999", t = 1000) == "5558"

assert Solution().smallestNumber(num = "11111", t = 26) == "-1"

assert Solution().smallestNumber(num = "1001", t = 1) == "1111"

assert Solution().smallestNumber(num = "9999", t = 9999) == "-1"

assert Solution().smallestNumber(num = "23456789", t = 987654) == "-1"

assert Solution().smallestNumber(num = "12355", t = 50) == "12355"

assert Solution().smallestNumber(num = "2468", t = 384) == "2468"

assert Solution().smallestNumber(num = "1111111111", t = 26) == "-1"

assert Solution().smallestNumber(num = "13579", t = 945) == "13579"

assert Solution().smallestNumber(num = "12305", t = 60) == "12325"

assert Solution().smallestNumber(num = "1000001", t = 125) == "1111555"

assert Solution().smallestNumber(num = "7654321", t = 123456) == "-1"

Answer: assert Solution().smallestNumber(num = "99999", t = 987654) == "-1"

Answer: assert Solution().smallestNumber(num = "1000", t = 10) == "1125"

Answer: assert Solution().smallestNumber(num = "123456789", t = 3628800) == "145578899"

Answer: assert Solution().smallestNumber(num = "101010", t = 10) == "111125"

Answer: assert Solution().smallestNumber(num = "202020", t = 20) == "211125"

Answer: assert Solution().smallestNumber(num = "1999", t = 2000) == "25558"

Answer: assert Solution().smallestNumber(num = "2222222222", t = 1024) == "2222222222"

Answer: assert Solution().smallestNumber(num = "987654321", t = 362880) == "987654321"

Answer: assert Solution().smallestNumber(num = "1234", t = 256) == "1488"

Answer: assert Solution().smallestNumber(num = "112233445566778899", t = 897654321) == "-1"

Answer: assert Solution().smallestNumber(num = "1001", t = 10) == "1125"

Answer: assert Solution().smallestNumber(num = "999", t = 1000) == "5558"

Answer: assert Solution().smallestNumber(num = "11111", t = 26) == "-1"

Answer: assert Solution().smallestNumber(num = "1001", t = 1) == "1111"

Answer: assert Solution().smallestNumber(num = "9999", t = 9999) == "-1"

Answer: assert Solution().smallestNumber(num = "987654321", t = 100) == "987654325"

Answer: assert Solution().smallestNumber(num = "23456789", t = 987654) == "-1"

Answer: assert Solution().smallestNumber(num = "12355", t = 50) == "12355"

Answer: assert Solution().smallestNumber(num = "876543219", t = 87654321) == "-1"

Answer: assert Solution().smallestNumber(num = "5555555555555555555555555555555555555555555555", t = 1125899906842624) == "5555555555555555555555555555584888888888888888"

Answer: assert Solution().smallestNumber(num = "111122223333444455556666777788889999", t = 1000000000) == "111122223333444455556666777788955555"

Answer: assert Solution().smallestNumber(num = "2468", t = 384) == "2468"

Answer: assert Solution().smallestNumber(num = "3141592653589793238462643383279", t = 600851475143) == "-1"

Answer: assert Solution().smallestNumber(num = "12345678901234567890", t = 362880000000000) == "12555555555578888899"

Answer: assert Solution().smallestNumber(num = "111111111111111111111111", t = 26) == "-1"

Answer: assert Solution().smallestNumber(num = "98765432109876543210", t = 1000000000000000000) == "555555555555555555888888"

Answer: assert Solution().smallestNumber(num = "10101010101010101010101010101010", t = 256) == "11111111111111111111111111111488"

Answer: assert Solution().smallestNumber(num = "222222222", t = 1024) == "222222224"

Answer: assert Solution().smallestNumber(num = "5555555555555555555555555555555", t = 15625) == "5555555555555555555555555555555"

Answer: assert Solution().smallestNumber(num = "99999999999999999999", t = 1000000000000000000) == "555555555555555555888888"

Answer: assert Solution().smallestNumber(num = "987654321234567898765432123456789", t = 9876543210987654321) == "-1"

Answer: assert Solution().smallestNumber(num = "3333333333", t = 116640000) == "4555588999"

Answer: assert Solution().smallestNumber(num = "1111111111", t = 100000000000) == "455555555555888"

Answer: assert Solution().smallestNumber(num = "99999999999999999999", t = 984150000000000) == "115555555555568889999"

Answer: assert Solution().smallestNumber(num = "12345678987654321234567898765432123456789", t = 283115500771200) == "-1"

Answer: assert Solution().smallestNumber(num = "1234567898765432123456789876543212345678987654321", t = 437893890384690340821512677536) == "-1"

Answer: assert Solution().smallestNumber(num = "11111111111111111111111111111111111111111111111", t = 3456789123456789123) == "-1"

Answer: assert Solution().smallestNumber(num = "9999999999", t = 1000000000000000) == "55555555555555588888"

Answer: assert Solution().smallestNumber(num = "1234567890", t = 362880) == "1234567891"

Answer: assert Solution().smallestNumber(num = "2222222222", t = 256) == "2222222222"

Answer: assert Solution().smallestNumber(num = "5432109876543210987654321", t = 123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "999999999999999999999999999999999999", t = 999999999999999999999999999999999999) == "-1"

Answer: assert Solution().smallestNumber(num = "44444444444444444444", t = 111111111111111111111) == "-1"

Answer: assert Solution().smallestNumber(num = "1020304050", t = 60) == "1111111256"

Answer: assert Solution().smallestNumber(num = "33333333333333333333", t = 243) == "33333333333333333333"

Answer: assert Solution().smallestNumber(num = "999999999999999999", t = 999999999999999) == "-1"

Answer: assert Solution().smallestNumber(num = "111111111", t = 362880) == "112578899"

Answer: assert Solution().smallestNumber(num = "59731846205973184620", t = 555555555555555555555) == "-1"

Answer: assert Solution().smallestNumber(num = "333333333", t = 729) == "333333333"

Answer: assert Solution().smallestNumber(num = "1111111111111111111111111111112", t = 720) == "1111111111111111111111111112589"

Answer: assert Solution().smallestNumber(num = "1000000000", t = 1) == "1111111111"

Answer: assert Solution().smallestNumber(num = "999999999", t = 100000000000000) == "4555555555555558888"

Answer: assert Solution().smallestNumber(num = "9999999999999999999999999999999", t = 1000000000000000000) == "11111111555555555555555555888888"

Answer: assert Solution().smallestNumber(num = "1111111111", t = 26) == "-1"

Answer: assert Solution().smallestNumber(num = "44444444444444444444", t = 65536) == "44444444444444444444"

Answer: assert Solution().smallestNumber(num = "9999999999", t = 100000000000000) == "4555555555555558888"

Answer: assert Solution().smallestNumber(num = "13579", t = 945) == "13579"

Answer: assert Solution().smallestNumber(num = "555555555555555555555555", t = 12345678901234567890) == "-1"

Answer: assert Solution().smallestNumber(num = "4444444444", t = 1048576) == "4444444444"

Answer: assert Solution().smallestNumber(num = "22222222222222222222", t = 12345678901234567890) == "-1"

Answer: assert Solution().smallestNumber(num = "66666666666666666666", t = 1296) == "66666666666666666666"

Answer: assert Solution().smallestNumber(num = "111111111111111111111111", t = 1234567890123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "98765432109876543210", t = 333333333333333333333) == "-1"

Answer: assert Solution().smallestNumber(num = "1999999999", t = 987654321) == "-1"

Answer: assert Solution().smallestNumber(num = "55555555555555555555", t = 15625) == "55555555555555555555"

Answer: assert Solution().smallestNumber(num = "10101010101010101010", t = 1000000) == "11111111111155555588"

Answer: assert Solution().smallestNumber(num = "666666666", t = 2176782336) == "8888999999"

Answer: assert Solution().smallestNumber(num = "1111111111", t = 1024) == "1111112888"

Answer: assert Solution().smallestNumber(num = "23456789", t = 123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "888888888888888888", t = 987654321) == "-1"

Answer: assert Solution().smallestNumber(num = "777777777", t = 5764801) == "777777777"

Answer: assert Solution().smallestNumber(num = "13579246801357924680", t = 56789012345678901234) == "-1"

Answer: assert Solution().smallestNumber(num = "7777777777", t = 5764801) == "7777777777"

Answer: assert Solution().smallestNumber(num = "5678901234", t = 123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "9876543210", t = 3628800) == "9876543225"

Answer: assert Solution().smallestNumber(num = "234567891011121314151617181920", t = 1234567890123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "77777777777777777777", t = 16807) == "77777777777777777777"

Answer: assert Solution().smallestNumber(num = "5555555555", t = 15625) == "5555555555"

Answer: assert Solution().smallestNumber(num = "9876543210", t = 1000) == "9876543255"

Answer: assert Solution().smallestNumber(num = "99999999999999999999", t = 99999999999999999999) == "-1"

Answer: assert Solution().smallestNumber(num = "112233445566778899", t = 444444444444444444444) == "-1"

Answer: assert Solution().smallestNumber(num = "3333333333", t = 243) == "3333333333"

Answer: assert Solution().smallestNumber(num = "87654321", t = 518400) == "87655689"

Answer: assert Solution().smallestNumber(num = "12345678912345678912345678912345", t = 86400) == "12345678912345678912345678912345"

Answer: assert Solution().smallestNumber(num = "12305", t = 60) == "12325"

Answer: assert Solution().smallestNumber(num = "22222222222222222222", t = 8192) == "22222222222222222222"

Answer: assert Solution().smallestNumber(num = "246813579", t = 123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "56789123456789123456", t = 222222222222222222222) == "-1"

Answer: assert Solution().smallestNumber(num = "1234567890", t = 123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "77777777777777777777777777777777", t = 49) == "77777777777777777777777777777777"

Answer: assert Solution().smallestNumber(num = "99999999999999999999999999999999", t = 9876543210987654321) == "-1"

Answer: assert Solution().smallestNumber(num = "8888888888", t = 512) == "8888888888"

Answer: assert Solution().smallestNumber(num = "111111111111111111", t = 123456789) == "-1"

Answer: assert Solution().smallestNumber(num = "555555555", t = 244140625) == "555555555555"

Answer: assert Solution().smallestNumber(num = "12345678901234567890", t = 1000000000000000) == "55555555555555588888"

Answer: assert Solution().smallestNumber(num = "123456789012345678901234567890", t = 10000000000000000000) == "123456815555555555555555558888"

Answer: assert Solution().smallestNumber(num = "88888888888888888888", t = 262144) == "88888888888888888888"

Answer: assert Solution().smallestNumber(num = "9999999999", t = 987654321) == "-1"

Answer: assert Solution().smallestNumber(num = "22222222222222222222222222222222", t = 100000000000000000) == "22222222222222555555555555555558"

Answer: assert Solution().smallestNumber(num = "999999999", t = 987654321) == "-1"

Answer: assert Solution().smallestNumber(num = "1357913579", t = 945) == "1357913579"

Answer: assert Solution().smallestNumber(num = "987654321", t = 3628800) == "987654345"

Answer: assert Solution().smallestNumber(num = "1357913579", t = 65536) == "1358128888"

Answer: assert Solution().smallestNumber(num = "112233445566778899", t = 100000000000000) == "4555555555555558888"

Answer: assert Solution().smallestNumber(num = "9876543210", t = 512000000) == "55555588888"

Answer: assert Solution().smallestNumber(num = "1234567891234567891234567891234", t = 120960000) == "1234567891234567891234567891235"

Answer: assert Solution().smallestNumber(num = "222222222", t = 4096) == "222222248"

Answer: assert Solution().smallestNumber(num = "12345678901234567890", t = 12345678901234567890) == "-1"

Answer: assert Solution().smallestNumber(num = "444444444", t = 1679616) == "444449999"

Answer: assert Solution().smallestNumber(num = "12345678901234567890", t = 9876543210) == "-1"

Answer: assert Solution().smallestNumber(num = "2333333333333333333333333333333", t = 1024) == "2333333333333333333333333333888"

Answer: assert Solution().smallestNumber(num = "37492653842783942378492378492384792834792384792384792384", t = 1000000000) == "37492653842783942378492378492384792834792384792455555555"

Answer: assert Solution().smallestNumber(num = "9876543210987654321", t = 50000000000000) == "12555555555555558888"

Answer: assert Solution().smallestNumber(num = "55555555555555555555555555555555", t = 125) == "55555555555555555555555555555555"

Answer: assert Solution().smallestNumber(num = "864208642086420", t = 4096) == "864211111111148"

Answer: assert Solution().smallestNumber(num = "123456789012345678901234567890", t = 987654321098765432109876543210) == "-1"

Answer: assert Solution().smallestNumber(num = "12345678910111213141516171819202122232425", t = 1111111111111111111) == "-1"

Answer: assert Solution().smallestNumber(num = "10203040506070809", t = 3628800) == "11111111145578899"

Answer: assert Solution().smallestNumber(num = "12345678901234567890", t = 280) == "12345678911111111111"

Answer: assert Solution().smallestNumber(num = "1010101010", t = 1000) == "1111115558"

Answer: assert Solution().smallestNumber(num = "1000001", t = 125) == "1111555"

Answer: assert Solution().smallestNumber(num = "88888888888888888888", t = 65536) == "88888888888888888888"

Answer: assert Solution().smallestNumber(num = "1111111111111111111111111111111", t = 26) == "-1"

Answer: assert Solution().smallestNumber(num = "987654321098765432109876543210", t = 1000000000000000) == "987654321111145555555555555588"

Answer: assert Solution().smallestNumber(num = "7654321", t = 123456) == "-1"

Answer: assert Solution().smallestNumber(num = "112233445566778899", t = 1234567890123456789) == "-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)

FACTOR_COUNTS = {
    0: collections.Counter(),
    1: collections.Counter(),
    2: collections.Counter([2]),
    3: collections.Counter([3]),
    4: collections.Counter([2, 2]),
    5: collections.Counter([5]),
    6: collections.Counter([2, 3]),
    7: collections.Counter([7]),
    8: collections.Counter([2, 2, 2]),
    9: collections.Counter([3, 3]),
}

class Solution:
  def smallestNumber(self, num: str, t: int) -> str:
    primeCount, isDivisible = self._getPrimeCount(t)
    if not isDivisible:
      return '-1'

    factorCount = self._getFactorCount(primeCount)
    if sum(factorCount.values()) > len(num):
      return ''.join(factor * freq for factor, freq in factorCount.items())

    primeCountPrefix = sum((FACTOR_COUNTS[int(c)]
                            for c in num), start=collections.Counter())
    firstZeroIndex = next((i for i, d in enumerate(num) if d == '0'), len(num))
    if firstZeroIndex == len(num) and primeCount <= primeCountPrefix:
      return num

    for i, c in reversed(list(enumerate(num))):
      d = int(c)
      # Remove the current digit's factors from primeCountPrefix.
      primeCountPrefix -= FACTOR_COUNTS[d]
      spaceAfterThisDigit = len(num) - 1 - i
      if i <= firstZeroIndex:
        for biggerDigit in range(d + 1, 10):
          # Compute the required factors after replacing with a larger digit.
          factorsAfterReplacement = self._getFactorCount(
              primeCount - primeCountPrefix - FACTOR_COUNTS[biggerDigit]
          )
          # Check if the replacement is possible within the available space.
          if sum(factorsAfterReplacement.values()) <= spaceAfterThisDigit:
            # Fill extra space with '1', if any, and construct the result.
            fillOnes = spaceAfterThisDigit - sum(
                factorsAfterReplacement.values())
            return (
                num[:i]  # Keep the prefix unchanged.
                + str(biggerDigit)  # Replace the current digit.
                + '1' * fillOnes  # Fill remaining space with '1'.
                + ''.join(factor * freq for factor,
                          freq in factorsAfterReplacement.items())
            )

    # No solution of the same length exists, so we need to extend the number
    # by prepending '1's and adding the required factors.
    factorCount = self._getFactorCount(primeCount)
    return (
        '1' * (len(num) + 1 - sum(factorCount.values()))
        + ''.join(factor * freq for factor, freq in factorCount.items())
    )

  def _getPrimeCount(self, t: int) -> tuple[dict[int, int], bool]:
    """
    Returns the count of prime factors of t and if t is divisible by 2, 3, 5, 7.
    """
    count = collections.Counter()
    for prime in [2, 3, 5, 7]:
      while t % prime == 0:
        t //= prime
        count[prime] += 1
    return count, t == 1

  def _getFactorCount(self, count: dict[int, int]) -> dict[str, int]:
    """Returns the required factors to form the smallest number."""
    count8, remaining2 = divmod(count[2], 3)  # 2^3 = 8
    count9, count3 = divmod(count[3], 2)  # 3^2 = 9
    count4, count2 = divmod(remaining2, 2)  # 2^2 = 4
    # Combine 2 and 3 to 6 if both are present.
    count2, count3, count6 = ((0, 0, 1) if count2 == 1 and count3 == 1
                              else (count2, count3, 0))
    # Combine 3 and 4 to 2 and 6 if both are present.
    count2, count6, count3, count4 = ((1, 1, 0, 0)
                                      if count3 == 1 and count4 == 1
                                      else (count2, count6, count3, count4))
    return {'2': count2, '3': count3, '4': count4, '5': count[5],
            '6': count6, '7': count[7], '8': count8, '9': count9}