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 n representing the dimensions of an n x n grid, with the origin at the bottom-left corner of the grid. You are also given a 2D array of coordinates rectangles, where rectangles[i] is in the form [startx, starty, endx, endy], representing a rectangle on the grid. Each rectangle is defined as follows:

(startx, starty): The bottom-left corner of the rectangle.
(endx, endy): The top-right corner of the rectangle.

Note that the rectangles do not overlap. Your task is to determine if it is possible to make either two horizontal or two vertical cuts on the grid such that:

Each of the three resulting sections formed by the cuts contains at least one rectangle.
Every rectangle belongs to exactly one section.

Return true if such cuts can be made; otherwise, return false.

Example 1:

Input: n = 5, rectangles = [[1,0,5,2],[0,2,2,4],[3,2,5,3],[0,4,4,5]]
Output: true
Explanation:

The grid is shown in the diagram. We can make horizontal cuts at y = 2 and y = 4. Hence, output is true.

Example 2:

Input: n = 4, rectangles = [[0,0,1,1],[2,0,3,4],[0,2,2,3],[3,0,4,3]]
Output: true
Explanation:

We can make vertical cuts at x = 2 and x = 3. Hence, output is true.

Example 3:

Input: n = 4, rectangles = [[0,2,2,4],[1,0,3,2],[2,2,3,4],[3,0,4,2],[3,2,4,4]]
Output: false
Explanation:
We cannot make two horizontal or two vertical cuts that satisfy the conditions. Hence, output is false.

Constraints:

3 <= n <= 109
3 <= rectangles.length <= 105
0 <= rectangles[i][0] < rectangles[i][2] <= n
0 <= rectangles[i][1] < rectangles[i][3] <= n
No two rectangles overlap.

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

SOLUTION:

Question:
PROBLEM:
You are given an integer n representing the dimensions of an n x n grid, with the origin at the bottom-left corner of the grid. You are also given a 2D array of coordinates rectangles, where rectangles[i] is in the form [startx, starty, endx, endy], representing a rectangle on the grid. Each rectangle is defined as follows:

(startx, starty): The bottom-left corner of the rectangle.
(endx, endy): The top-right corner of the rectangle.

Note that the rectangles do not overlap. Your task is to determine if it is possible to make either two horizontal or two vertical cuts on the grid such that:

Each of the three resulting sections formed by the cuts contains at least one rectangle.
Every rectangle belongs to exactly one section.

Return true if such cuts can be made; otherwise, return false.

Example 1:

Input: n = 5, rectangles = [[1,0,5,2],[0,2,2,4],[3,2,5,3],[0,4,4,5]]
Output: true
Explanation:

The grid is shown in the diagram. We can make horizontal cuts at y = 2 and y = 4. Hence, output is true.

Example 2:

Input: n = 4, rectangles = [[0,0,1,1],[2,0,3,4],[0,2,2,3],[3,0,4,3]]
Output: true
Explanation:

We can make vertical cuts at x = 2 and x = 3. Hence, output is true.

Example 3:

Input: n = 4, rectangles = [[0,2,2,4],[1,0,3,2],[2,2,3,4],[3,0,4,2],[3,2,4,4]]
Output: false
Explanation:
We cannot make two horizontal or two vertical cuts that satisfy the conditions. Hence, output is false.

Constraints:

3 <= n <= 109
3 <= rectangles.length <= 105
0 <= rectangles[i][0] < rectangles[i][2] <= n
0 <= rectangles[i][1] < rectangles[i][3] <= n
No two rectangles overlap.

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

SOLUTION:

Answer: assert Solution().checkValidCuts(n = 5, rectangles = [[1,0,5,2],[0,2,2,4],[3,2,5,3],[0,4,4,5]]) == True

Answer: assert Solution().checkValidCuts(n = 4, rectangles = [[0,0,1,1],[2,0,3,4],[0,2,2,3],[3,0,4,3]]) == True

Answer: assert Solution().checkValidCuts(n = 4, rectangles = [[0,2,2,4],[1,0,3,2],[2,2,3,4],[3,0,4,2],[3,2,4,4]]) == False

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[0,6,3,9],[3,6,6,9],[6,6,9,9]]) == True

Answer: assert Solution().checkValidCuts(n = 14, rectangles = [[1,1,5,5],[6,1,10,5],[11,1,14,5],[1,6,5,10],[6,6,10,10],[11,6,14,10],[1,11,5,14],[6,11,10,14],[11,11,14,14]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,2,3],[2,0,4,3],[4,0,6,3],[6,0,8,3],[8,0,10,3],[1,4,3,6],[3,4,5,6],[5,4,7,6],[7,4,9,6],[0,7,2,10],[2,7,4,10],[4,7,6,10],[6,7,8,10],[8,7,10,10]]) == True

Answer: assert Solution().checkValidCuts(n = 20, rectangles = [[0,0,5,5],[5,0,10,5],[10,0,15,5],[15,0,20,5],[0,5,5,10],[5,5,10,10],[10,5,15,10],[15,5,20,10],[0,10,5,15],[5,10,10,15],[10,10,15,15],[15,10,20,15],[0,15,5,20],[5,15,10,20],[10,15,15,20],[15,15,20,20]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,4,3],[4,0,7,2],[7,0,10,1],[0,3,3,7],[3,3,10,5],[0,7,5,10],[5,7,8,8],[8,7,10,10]]) == True

Answer: assert Solution().checkValidCuts(n = 15, rectangles = [[0,0,5,4],[5,0,10,3],[10,0,15,2],[0,4,3,8],[3,4,7,6],[7,4,11,5],[11,4,15,7],[0,8,5,10],[5,8,10,9],[10,8,15,12],[0,10,7,12],[7,10,11,11],[11,10,15,15]]) == True

Answer: assert Solution().checkValidCuts(n = 8, rectangles = [[0,0,3,2],[3,0,5,3],[5,0,8,2],[0,2,3,6],[3,3,5,6],[5,2,8,6],[0,6,3,8],[3,6,5,8],[5,6,8,8]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,3,3],[3,0,6,6],[6,0,10,10],[0,3,3,6],[3,3,6,9],[6,3,10,10],[0,6,3,10],[3,6,6,10],[6,6,10,10]]) == True

Answer: assert Solution().checkValidCuts(n = 30, rectangles = [[0,0,10,10],[10,0,20,10],[20,0,30,10],[0,10,10,20],[10,10,20,20],[20,10,30,20],[0,20,10,30],[10,20,20,30],[20,20,30,30],[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]]) == True

Answer: assert Solution().checkValidCuts(n = 11, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[9,0,11,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[9,3,11,6],[0,6,3,9],[3,6,6,9],[6,6,9,9],[9,6,11,9]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[1,1,3,3],[3,1,5,3],[5,1,7,3],[7,1,9,3],[1,3,3,5],[3,3,5,5],[5,3,7,5],[7,3,9,5],[1,5,3,7],[3,5,5,7],[5,5,7,7],[7,5,9,7],[1,7,3,9],[3,7,5,9],[5,7,7,9],[7,7,9,9]]) == True

Answer: assert Solution().checkValidCuts(n = 7, rectangles = [[0,0,1,1],[1,0,2,1],[2,0,3,1],[3,0,4,1],[4,0,5,1],[5,0,6,1],[6,0,7,1],[0,1,1,2],[1,1,2,2],[2,1,3,2],[3,1,4,2],[4,1,5,2],[5,1,6,2],[6,1,7,2],[0,2,1,3],[1,2,2,3],[2,2,3,3],[3,2,4,3],[4,2,5,3],[5,2,6,3],[6,2,7,3],[0,3,1,4],[1,3,2,4],[2,3,3,4],[3,3,4,4],[4,3,5,4],[5,3,6,4],[6,3,7,4],[0,4,1,5],[1,4,2,5],[2,4,3,5],[3,4,4,5],[4,4,5,5],[5,4,6,5],[6,4,7,5]]) == True

Answer: assert Solution().checkValidCuts(n = 9, rectangles = [[0,0,3,1],[3,0,6,1],[6,0,9,1],[0,1,3,2],[3,1,6,2],[6,1,9,2],[0,2,3,3],[3,2,6,3],[6,2,9,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[0,6,3,7],[3,6,6,7],[6,6,9,7],[0,7,3,9],[3,7,6,9],[6,7,9,9]]) == True

Answer: assert Solution().checkValidCuts(n = 8, rectangles = [[0,0,2,2],[2,0,4,2],[4,0,6,2],[6,0,8,2],[0,2,2,4],[2,2,4,4],[4,2,6,4],[6,2,8,4],[0,4,2,6],[2,4,4,6],[4,4,6,6],[6,4,8,6],[0,6,2,8],[2,6,4,8],[4,6,6,8],[6,6,8,8]]) == True

Answer: assert Solution().checkValidCuts(n = 8, rectangles = [[0,0,2,1],[2,0,4,1],[4,0,6,1],[6,0,8,1],[0,1,2,2],[2,1,4,2],[4,1,6,2],[6,1,8,2],[0,2,2,3],[2,2,4,3],[4,2,6,3],[6,2,8,3],[0,3,2,4],[2,3,4,4],[4,3,6,4],[6,3,8,4],[0,4,2,5],[2,4,4,5],[4,4,6,5],[6,4,8,5],[0,5,2,6],[2,5,4,6],[4,5,6,6],[6,5,8,6],[0,6,2,7],[2,6,4,7],[4,6,6,7],[6,6,8,7],[0,7,2,8],[2,7,4,8],[4,7,6,8],[6,7,8,8]]) == True

Answer: assert Solution().checkValidCuts(n = 7, rectangles = [[0,0,1,1],[1,0,2,1],[2,0,3,1],[3,0,4,1],[4,0,5,1],[5,0,6,1],[6,0,7,1],[0,1,1,2],[1,1,2,2],[2,1,3,2],[3,1,4,2],[4,1,5,2],[5,1,6,2],[6,1,7,2],[0,2,1,3],[1,2,2,3],[2,2,3,3],[3,2,4,3],[4,2,5,3],[5,2,6,3],[6,2,7,3]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[9,0,10,3],[0,3,10,6],[0,6,10,9],[0,9,10,10]]) == True

Answer: assert Solution().checkValidCuts(n = 20, rectangles = [[0,0,4,4],[4,0,8,4],[8,0,12,4],[12,0,16,4],[16,0,20,4],[0,4,4,8],[4,4,8,8],[8,4,12,8],[12,4,16,8],[16,4,20,8],[0,8,4,12],[4,8,8,12],[8,8,12,12],[12,8,16,12],[16,8,20,12],[0,12,4,16],[4,12,8,16],[8,12,12,16],[12,12,16,16],[16,12,20,16],[0,16,8,20],[8,16,16,20],[16,16,20,20]]) == True

Answer: assert Solution().checkValidCuts(n = 20, rectangles = [[1,1,4,4],[5,5,8,8],[9,9,12,12],[13,13,16,16],[17,17,20,20],[1,5,4,8],[5,9,8,12],[9,13,12,16],[13,17,16,20]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,4,3],[4,0,7,2],[7,0,10,4],[1,3,5,6],[5,3,8,5],[8,3,10,10],[0,6,4,9],[4,6,8,8],[8,6,10,10]]) == False

Answer: assert Solution().checkValidCuts(n = 15, rectangles = [[1,1,5,5],[6,1,10,5],[11,1,15,5],[1,6,5,10],[6,6,10,10],[11,6,15,10],[1,11,5,15],[6,11,10,15],[11,11,15,15]]) == True

Answer: assert Solution().checkValidCuts(n = 11, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[9,0,11,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[9,3,11,6],[0,6,3,9],[3,6,6,9],[6,6,9,9],[9,6,11,9],[0,9,3,11],[3,9,6,11],[6,9,9,11],[9,9,11,11]]) == True

Answer: assert Solution().checkValidCuts(n = 15, rectangles = [[0,0,5,3],[5,0,10,3],[10,0,15,3],[0,3,5,6],[5,3,10,6],[10,3,15,6],[0,6,5,9],[5,6,10,9],[10,6,15,9],[0,9,5,12],[5,9,10,12],[10,9,15,12],[0,12,5,15],[5,12,10,15],[10,12,15,15]]) == True

Answer: assert Solution().checkValidCuts(n = 8, rectangles = [[0,0,2,2],[2,0,4,4],[4,0,6,6],[6,0,8,8],[0,2,2,4],[2,2,4,6],[4,2,6,8],[6,2,8,8],[0,4,2,6],[2,4,4,8],[4,4,6,8],[6,4,8,8],[0,6,2,8],[2,6,4,8],[4,6,6,8],[6,6,8,8]]) == True

Answer: assert Solution().checkValidCuts(n = 6, rectangles = [[0,0,2,1],[2,0,4,1],[4,0,6,1],[0,1,2,2],[2,1,4,2],[4,1,6,2],[0,2,2,3],[2,2,4,3],[4,2,6,3],[0,3,2,4],[2,3,4,4],[4,3,6,4],[0,4,2,5],[2,4,4,5],[4,4,6,5],[0,5,2,6],[2,5,4,6],[4,5,6,6]]) == True

Answer: assert Solution().checkValidCuts(n = 7, rectangles = [[0,0,1,2],[1,0,2,2],[2,0,3,2],[3,0,4,2],[4,0,5,2],[5,0,6,2],[6,0,7,2],[0,2,1,4],[1,2,2,4],[2,2,3,4],[3,2,4,4],[4,2,5,4],[5,2,6,4],[6,2,7,4],[0,4,1,6],[1,4,2,6],[2,4,3,6],[3,4,4,6],[4,4,5,6],[5,4,6,6],[6,4,7,6],[0,6,1,7],[1,6,2,7],[2,6,3,7],[3,6,4,7],[4,6,5,7],[5,6,6,7],[6,6,7,7]]) == True

Answer: assert Solution().checkValidCuts(n = 9, rectangles = [[1,1,4,4],[4,1,7,4],[7,1,9,4],[1,4,4,7],[4,4,7,7],[7,4,9,7],[1,7,4,9],[4,7,7,9],[7,7,9,9]]) == True

Answer: assert Solution().checkValidCuts(n = 15, rectangles = [[0,0,4,5],[4,0,8,4],[8,0,12,6],[12,0,15,3],[0,5,4,10],[4,4,8,8],[8,6,12,10],[12,3,15,7],[0,10,4,15],[4,8,8,12],[8,10,12,15],[12,7,15,10]]) == True

Answer: assert Solution().checkValidCuts(n = 25, rectangles = [[0,0,7,6],[7,0,14,5],[14,0,21,7],[21,0,25,3],[0,6,7,12],[7,5,14,10],[14,7,21,12],[21,3,25,8],[0,12,7,18],[7,10,14,15],[14,12,21,18],[21,8,25,13],[0,18,7,25],[7,15,14,20],[14,18,21,25],[21,13,25,18]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,4,4],[5,0,8,4],[0,5,3,9],[7,5,10,9]]) == False

Answer: assert Solution().checkValidCuts(n = 15, rectangles = [[0,0,5,5],[5,0,10,5],[10,0,15,5],[0,5,5,10],[5,5,10,10],[10,5,15,10],[0,10,5,15],[5,10,10,15],[10,10,15,15],[2,2,3,3],[7,7,8,8],[12,12,13,13]]) == True

Answer: assert Solution().checkValidCuts(n = 15, rectangles = [[0,0,5,5],[5,0,10,5],[10,0,15,5],[0,5,5,10],[5,5,10,10],[10,5,15,10],[0,10,5,15],[5,10,10,15],[10,10,15,15]]) == True

Answer: assert Solution().checkValidCuts(n = 15, rectangles = [[1,1,4,4],[5,1,8,4],[9,1,12,4],[13,1,15,4],[1,5,4,8],[5,5,8,8],[9,5,12,8],[13,5,15,8],[1,9,4,12],[5,9,8,12],[9,9,12,12],[13,9,15,12],[1,13,4,15],[5,13,8,15],[9,13,12,15],[13,13,15,15]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,2,3],[3,0,5,4],[6,0,8,2],[0,4,3,7],[3,4,5,6],[6,4,8,7],[0,7,3,10],[3,7,5,10],[6,7,8,10]]) == True

Answer: assert Solution().checkValidCuts(n = 7, rectangles = [[0,0,2,2],[2,0,4,2],[4,0,6,2],[6,0,7,2],[0,2,2,4],[2,2,4,4],[4,2,6,4],[6,2,7,4],[0,4,2,6],[2,4,4,6],[4,4,6,6],[6,4,7,6],[0,6,2,7],[2,6,4,7],[4,6,6,7],[6,6,7,7]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,3,3],[4,0,6,3],[7,0,10,3],[0,3,3,6],[4,3,6,6],[7,3,10,6],[0,6,3,9],[4,6,6,9],[7,6,10,9]]) == True

Answer: assert Solution().checkValidCuts(n = 9, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[0,6,3,9],[3,6,6,9],[6,6,9,9],[1,1,2,2],[2,1,3,2],[4,1,5,2],[5,1,6,2],[7,1,8,2],[8,1,9,2],[1,4,2,5],[2,4,3,5],[4,4,5,5],[5,4,6,5],[7,4,8,5],[8,4,9,5],[1,7,2,8],[2,7,3,8],[4,7,5,8],[5,7,6,8],[7,7,8,8],[8,7,9,8]]) == True

Answer: assert Solution().checkValidCuts(n = 9, rectangles = [[1,1,2,2],[2,1,3,2],[3,1,4,2],[1,2,2,3],[2,2,3,3],[3,2,4,3],[1,3,2,4],[2,3,3,4],[3,3,4,4],[5,5,6,6],[6,5,7,6],[7,5,8,6],[5,6,6,7],[6,6,7,7],[7,6,8,7],[5,7,6,8],[6,7,7,8],[7,7,8,8]]) == True

Answer: assert Solution().checkValidCuts(n = 12, rectangles = [[0,0,4,4],[4,0,8,4],[8,0,12,4],[0,4,4,8],[4,4,8,8],[8,4,12,8],[0,8,4,12],[4,8,8,12],[8,8,12,12]]) == True

Answer: assert Solution().checkValidCuts(n = 12, rectangles = [[1,1,3,3],[4,1,6,3],[7,1,9,3],[10,1,12,3],[1,4,3,6],[4,4,6,6],[7,4,9,6],[10,4,12,6],[1,7,3,9],[4,7,6,9],[7,7,9,9],[10,7,12,9],[1,10,3,12],[4,10,6,12],[7,10,9,12],[10,10,12,12]]) == True

Answer: assert Solution().checkValidCuts(n = 8, rectangles = [[0,0,2,2],[0,2,2,4],[0,4,2,6],[0,6,2,8],[2,0,4,2],[2,2,4,4],[2,4,4,6],[2,6,4,8],[4,0,6,2],[4,2,6,4],[4,4,6,6],[4,6,6,8],[6,0,8,2],[6,2,8,4],[6,4,8,6],[6,6,8,8]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,2,2],[2,2,4,4],[4,4,6,6],[6,6,8,8],[8,8,10,10]]) == True

Answer: assert Solution().checkValidCuts(n = 6, rectangles = [[1,1,2,2],[2,1,3,2],[3,1,4,2],[4,1,5,2],[5,1,6,2],[1,2,2,3],[2,2,3,3],[3,2,4,3],[4,2,5,3],[5,2,6,3],[1,3,2,4],[2,3,3,4],[3,3,4,4],[4,3,5,4],[5,3,6,4],[1,4,2,5],[2,4,3,5],[3,4,4,5],[4,4,5,5],[5,4,6,5]]) == True

Answer: assert Solution().checkValidCuts(n = 12, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[9,0,12,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[9,3,12,6],[0,6,3,9],[3,6,6,9],[6,6,9,9],[9,6,12,9],[0,9,3,12],[3,9,6,12],[6,9,9,12],[9,9,12,12]]) == True

Answer: assert Solution().checkValidCuts(n = 9, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[0,6,3,9],[3,6,6,9],[6,6,9,9]]) == True

Answer: assert Solution().checkValidCuts(n = 6, rectangles = [[0,0,2,2],[2,2,4,4],[4,0,6,2],[0,4,2,6],[2,4,4,6],[4,4,6,6]]) == True

Answer: assert Solution().checkValidCuts(n = 9, rectangles = [[0,0,2,2],[2,0,4,2],[4,0,6,2],[6,0,8,2],[8,0,9,2],[0,2,1,4],[1,2,3,4],[3,2,5,4],[5,2,7,4],[7,2,9,4],[0,4,2,6],[2,4,4,6],[4,4,6,6],[6,4,8,6],[8,4,9,6],[0,6,2,8],[2,6,4,8],[4,6,6,8],[6,6,8,8],[8,6,9,8],[0,8,2,9],[2,8,4,9],[4,8,6,9],[6,8,8,9],[8,8,9,9]]) == True

Answer: assert Solution().checkValidCuts(n = 8, rectangles = [[0,0,2,4],[2,0,4,4],[4,0,6,4],[6,0,8,4],[0,4,2,8],[2,4,4,8],[4,4,6,8],[6,4,8,8]]) == True

Answer: assert Solution().checkValidCuts(n = 10, rectangles = [[0,0,5,5],[5,0,10,5],[0,5,5,10],[5,5,10,10],[2,2,3,3],[7,7,8,8]]) == False

Answer: assert Solution().checkValidCuts(n = 12, rectangles = [[0,0,3,3],[3,0,6,3],[6,0,9,3],[9,0,12,3],[0,3,3,6],[3,3,6,6],[6,3,9,6],[9,3,12,6],[0,6,3,9],[3,6,6,9],[6,6,9,9],[9,6,12,9]]) == True

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 countLineIntersections(self, coordinates: List[tuple[int, int]]) -> bool:
        lines = 0
        overlap = 0
        for value, marker in coordinates:
            if marker == 0:
                overlap -= 1
            else:
                overlap += 1

            if overlap == 0:
                lines += 1

        return lines >= 3

    def checkValidCuts(self, n: int, rectangles: List[List[int]]) -> bool:
        y_coordinates = []
        x_coordinates = []

        for rect in rectangles:
            x1, y1, x2, y2 = rect
            y_coordinates.append((y1, 1))  # start
            y_coordinates.append((y2, 0))  # end

            x_coordinates.append((x1, 1))  # start
            x_coordinates.append((x2, 0))  # end

        # Sort by coordinate value, and for tie, put end (0) before start (1)
        y_coordinates.sort(key=lambda x: (x[0], x[1]))
        x_coordinates.sort(key=lambda x: (x[0], x[1]))

        return self.countLineIntersections(
            y_coordinates
        ) or self.countLineIntersections(x_coordinates)


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 an integer array nums, find the number of subsequences of size 5 of nums with a unique middle mode.
Since the answer may be very large, return it modulo 109 + 7.
A mode of a sequence of numbers is defined as the element that appears the maximum number of times in the sequence.
A sequence of numbers contains a unique mode if it has only one mode.
A sequence of numbers seq of size 5 contains a unique middle mode if the middle element (seq[2]) is a unique mode.

Example 1:

Input: nums = [1,1,1,1,1,1]
Output: 6
Explanation:
[1, 1, 1, 1, 1] is the only subsequence of size 5 that can be formed, and it has a unique middle mode of 1. This subsequence can be formed in 6 different ways, so the output is 6.

Example 2:

Input: nums = [1,2,2,3,3,4]
Output: 4
Explanation:
[1, 2, 2, 3, 4] and [1, 2, 3, 3, 4] each have a unique middle mode because the number at index 2 has the greatest frequency in the subsequence. [1, 2, 2, 3, 3] does not have a unique middle mode because 2 and 3 appear twice.

Example 3:

Input: nums = [0,1,2,3,4,5,6,7,8]
Output: 0
Explanation:
There is no subsequence of length 5 with a unique middle mode.

Constraints:

5 <= nums.length <= 1000
-109 <= nums[i] <= 109

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

SOLUTION:

Question:
PROBLEM:
Given an integer array nums, find the number of subsequences of size 5 of nums with a unique middle mode.
Since the answer may be very large, return it modulo 109 + 7.
A mode of a sequence of numbers is defined as the element that appears the maximum number of times in the sequence.
A sequence of numbers contains a unique mode if it has only one mode.
A sequence of numbers seq of size 5 contains a unique middle mode if the middle element (seq[2]) is a unique mode.

Example 1:

Input: nums = [1,1,1,1,1,1]
Output: 6
Explanation:
[1, 1, 1, 1, 1] is the only subsequence of size 5 that can be formed, and it has a unique middle mode of 1. This subsequence can be formed in 6 different ways, so the output is 6.

Example 2:

Input: nums = [1,2,2,3,3,4]
Output: 4
Explanation:
[1, 2, 2, 3, 4] and [1, 2, 3, 3, 4] each have a unique middle mode because the number at index 2 has the greatest frequency in the subsequence. [1, 2, 2, 3, 3] does not have a unique middle mode because 2 and 3 appear twice.

Example 3:

Input: nums = [0,1,2,3,4,5,6,7,8]
Output: 0
Explanation:
There is no subsequence of length 5 with a unique middle mode.

Constraints:

5 <= nums.length <= 1000
-109 <= nums[i] <= 109

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

SOLUTION:

assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,3,3,4,4,4,4,4]) == 951

assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,1,1,2,2,3,3]) == 221

assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,3,4,4,4,5,5,5,5]) == 1185

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

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

assert Solution().subsequencesWithMiddleMode(nums = [-1,-1,-1,0,0,1,1,1,2,2,2]) == 165

assert Solution().subsequencesWithMiddleMode(nums = [0,1,2,3,4,5,6,7,8]) == 0

assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,3,3,4,4,5,5]) == 80

assert Solution().subsequencesWithMiddleMode(nums = [10,20,30,40,50,60,70,80,90,100]) == 0

assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5]) == 0

assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5,1,2,3,4,5]) == 50

assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,3,3,3,3,3]) == 402

assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,5,5,5,5]) == 126

assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,3,3,3,3,3]) == 1503

assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,3,3,3,4,4,4,5,5,5]) == 1140

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

assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,5,5,5,5,5]) == 252

assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 3003

assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1]) == 6

assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5,1,2,3,4,5,1,2,3,4,5]) == 760

assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,4,4,4,4]) == 191

assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,3,4,4,4,4,5,5,5,5,5]) == 1409

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1]) == 5613

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1000000000,1000000000,1000000000,1000000000,1000000000]) == 1

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,3,3,4,4,4,4,4]) == 951

Answer: assert Solution().subsequencesWithMiddleMode(nums = [10,20,20,20,30,30,40,40,40,40,50,50,50,50,50]) == 1349

Answer: assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,1,1,2,2,3,3]) == 221

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,3,4,4,4,5,5,5,5]) == 1185

Answer: assert Solution().subsequencesWithMiddleMode(nums = [-1,-2,-3,-4,-5,-1,-2,-3,-4,-5,-1,-2,-3,-4,-5]) == 760

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

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

Answer: assert Solution().subsequencesWithMiddleMode(nums = [-1,-1,-1,0,0,1,1,1,2,2,2]) == 165

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

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,3,3,4,4,5,5]) == 80

Answer: assert Solution().subsequencesWithMiddleMode(nums = [10,20,30,40,50,60,70,80,90,100]) == 0

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5]) == 0

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5,1,2,3,4,5]) == 50

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5]) == 8799

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,3,3,3,3,3]) == 402

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7]) == 7500

Answer: assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,5,5,5,5]) == 126

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,3,3,3,3,3]) == 1503

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,3,3,3,4,4,4,5,5,5]) == 1140

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

Answer: assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,5,5,5,5,5]) == 252

Answer: assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 3003

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1]) == 6

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4]) == 12723

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5,1,2,3,4,5,1,2,3,4,5]) == 760

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

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,4,4,4,4]) == 191

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9]) == 495068

Answer: assert Solution().subsequencesWithMiddleMode(nums = [-5,-5,-5,-5,-5,-5,-5,-5,-5,-4,-4,-4,-4,-4,-4,-4,-4,-4,-3,-3,-3,-3,-3,-3,-3,-3,-3,-2,-2,-2,-2,-2,-2,-2,-2,-2]) == 184536

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4]) == 1207062

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 800874

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,2,2,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,7,7,7,8,8,8]) == 61089

Answer: assert Solution().subsequencesWithMiddleMode(nums = [-1,-1,-1,-1,-1,0,0,0,0,0,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3]) == 22505

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,17,17,17,18,18,18,19,19,19,20,20,20]) == 708453

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10,11,11,12,12,13,13,14,14,15,15]) == 18928

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8]) == 86599

Answer: assert Solution().subsequencesWithMiddleMode(nums = [10,20,20,30,30,30,40,40,40,40,50,50,50,50,50,60,60,60,60,60,60]) == 9107

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4]) == 184536

Answer: assert Solution().subsequencesWithMiddleMode(nums = [5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 27285336

Answer: assert Solution().subsequencesWithMiddleMode(nums = [0,0,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4]) == 727516

Answer: assert Solution().subsequencesWithMiddleMode(nums = [100,200,200,200,300,300,300,300,400,400,400,400,400,500,500,500,500,500,500,500,500,500,500]) == 18556

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9]) == 525765

Answer: assert Solution().subsequencesWithMiddleMode(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().subsequencesWithMiddleMode(nums = [1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5,1,2,3,4,5]) == 657990

Answer: assert Solution().subsequencesWithMiddleMode(nums = [10,20,20,20,30,30,30,30,30,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40,40]) == 104255

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3]) == 112113

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6]) == 86817

Answer: assert Solution().subsequencesWithMiddleMode(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]) == 37530

Answer: assert Solution().subsequencesWithMiddleMode(nums = [-1,-1,-1,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3]) == 190065

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9]) == 3885489

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9]) == 656149

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 2278561

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4]) == 85638

Answer: assert Solution().subsequencesWithMiddleMode(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]) == 21840

Answer: assert Solution().subsequencesWithMiddleMode(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]) == 0

Answer: assert Solution().subsequencesWithMiddleMode(nums = [100,100,100,100,100,100,101,101,101,101,101,102,102,102,102,102,102,103,103,103,103,103,103,103]) == 19524

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3]) == 127869

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9]) == 140000

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5]) == 348457

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,6]) == 85182

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1000000000,1000000000,1000000000,1000000000,1000000000,1,2,3,4,5]) == 126

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5]) == 30199

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,3,3,4,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]) == 8948

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5]) == 107588

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8]) == 55871

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10]) == 626260

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5]) == 160013

Answer: assert Solution().subsequencesWithMiddleMode(nums = [0,0,0,0,0,1,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8]) == 39039

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8]) == 63891

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2]) == 45738

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6]) == 158764

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7]) == 431134

Answer: assert Solution().subsequencesWithMiddleMode(nums = [7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9]) == 446508

Answer: assert Solution().subsequencesWithMiddleMode(nums = [0,0,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5]) == 105672

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9,10,10,10,10,10,10,10]) == 2700054

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5]) == 148228

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 544618

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5]) == 89038

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5]) == 82236

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10]) == 149526

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3]) == 98136

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9]) == 81604

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9,10,10,10,10,10,11,11,11,11,11,12,12,12,12,12,13,13,13,13,13,14,14,14,14,14]) == 2798264

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 580164

Answer: assert Solution().subsequencesWithMiddleMode(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]) == 61830

Answer: assert Solution().subsequencesWithMiddleMode(nums = [-1000000000,-1000000000,-1000000000,-1000000000,-1000000000,-1000000000,-1000000000,-1000000000,-1000000000,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2]) == 45738

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5]) == 378931

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4]) == 108611

Answer: assert Solution().subsequencesWithMiddleMode(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]) == 11440

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,3,4,4,4,4,5,5,5,5,5]) == 1409

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5,6,7,8,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10]) == 31668

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9]) == 443663

Answer: assert Solution().subsequencesWithMiddleMode(nums = [10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10]) == 169911

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4]) == 540508

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10]) == 208198

Answer: assert Solution().subsequencesWithMiddleMode(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]) == 0

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2]) == 2598960

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7]) == 739355

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5]) == 63770

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2]) == 169911

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60]) == 0

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8]) == 422646

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,11,11,11,12,12,12,13,13,13,14,14,14,15,15,15,16,16,16,17,17,17,18,18,18,19,19,19,20,20,20,21,21,21,22,22,22,23,23,23,24,24,24]) == 1777716

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,7]) == 719651

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9,9,9]) == 1026054

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 775155

Answer: assert Solution().subsequencesWithMiddleMode(nums = [9,8,8,7,7,7,6,6,6,6,5,5,5,5,5,4,4,4,4,4,4,3,3,3,3,3,3,3,2,2,2,2]) == 86599

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6,7,7,7,7,7,8,8,8,8,8,9,9,9,9,9]) == 403014

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10]) == 50392

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9]) == 48201

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,9,9,9,9]) == 455652

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4]) == 955914

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]) == 142506

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,2,3,3,3,3,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9]) == 600008

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5]) == 358897

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3]) == 102063

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2]) == 169911

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9]) == 33885

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3]) == 135996

Answer: assert Solution().subsequencesWithMiddleMode(nums = [10,10,10,10,10,10,10,10,10,10,9,9,9,9,9,9,9,9,9,8,8,8,8,8,8,8,8,8,7,7,7,7,7,7,7,7,7,6,6,6,6,6,6,6,6,6,5,5,5,5,5,5,5,5,5,4,4,4,4,4,4,4,4,4]) == 2927808

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7,7,7,7,8,8,8,8,9,9,9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,14,14,14,15,15,15,15,16,16,16,16,17,17,17,17,18,18,18,18]) == 2515320

Answer: assert Solution().subsequencesWithMiddleMode(nums = [-1,-1,-1,0,0,0,1,1,1,2,2,2,2,2,3]) == 1428

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]) == 2349060

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,2,2,2,3,3,3,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6]) == 367473

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,9]) == 2318085

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6]) == 73447

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8]) == 10375

Answer: assert Solution().subsequencesWithMiddleMode(nums = [10,10,10,10,10,10,10,10,10,10,20,20,20,20,20,20,20,20,20,20,20,30,30,30,30,30,30,30,30,30,30,30]) == 116676

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,2,2,2,2,2,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5]) == 224666

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,1,1,2,2,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9]) == 675066

Answer: assert Solution().subsequencesWithMiddleMode(nums = [1,1,2,2,2,3,3,3,3,4,4,4,5,5,5,6,6,6,7,7,7,8,8,8,9,9,9,10,10,10,10]) == 45852

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 __init__(self):
    self.MOD = 1_000_000_007

  def subsequencesWithMiddleMode(self, nums: list[int]) -> int:
    n = len(nums)
    ans = 0
    left = collections.Counter()
    right = collections.Counter()

    for i in range(2):
      left[nums[i]] += 1

    for i in range(2, n):
      right[nums[i]] += 1

    for i in range(2, n - 2):
      num = nums[i]
      right[num] -= 1
      if right[num] == 0:
        del right[num]

      leftCount = left[num]
      rightCount = right[num]
      leftOther = i - leftCount
      rightOther = n - 1 - i - rightCount

      # count[mode] = 5 -- [a a] a [a a]
      ans += math.comb(leftCount, 2) * math.comb(rightCount, 2)

      # count[mode] = 4 -- [a a] a [a ?]
      ans += math.comb(leftCount, 2) * rightCount * rightOther

      # count[mode] = 4 -- [a ?] a [a a]
      ans += leftCount * leftOther * math.comb(rightCount, 2)

      # count[mode] = 3 -- [a a] a [? ?]
      ans += math.comb(leftCount, 2) * math.comb(rightOther, 2)

      # count[mode] = 3 -- [? ?] a [a a]
      ans += math.comb(leftOther, 2) * math.comb(rightCount, 2)

      # count[mode] = 3 -- [a ?] a [a ?]
      ans += leftCount * leftOther * rightCount * rightOther

      # count[mode] = 2 -- [a ?] a [? ?]
      ans += leftCount * self._calc(num, leftOther, rightOther, left, right)

      # count[mode] = 2 -- [? ?] a [a ?]
      ans += rightCount * self._calc(num, rightOther, leftOther, right, left)

      ans %= self.MOD
      left[num] += 1

    return ans

  def _calc(
      self,
      a: int,
      other1: int,
      other2: int,
      count1: dict[int, int],
      count2: dict[int, int]
  ) -> int:
    """
    Returns the count of subsequences that have `a` as the middle number, where
    invalid subsequences are excluded.
    """
    # [a ?] a [? ?]
    res = (other1 * math.comb(other2, 2)) % self.MOD

    for b, b1 in count1.items():
      if b == a:
        continue
      b2 = count2[b]
      # Exclude triples -- [a b] a [b b].
      res = (res - b1 * math.comb(b2, 2)) % self.MOD
      # Exclude doubles -- [a b] a [b ?].
      res = (res - b1 * b2 * (other2 - b2)) % self.MOD

    for b, b2 in count2.items():
      if b == a:
        continue
      b1 = count1[b]
      # Exclude doubles -- [a ?] a [b b].
      res = (res - (other1 - b1) * math.comb(b2, 2)) % self.MOD

    return (res + self.MOD) % self.MOD


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 an integer k.
You are allowed to perform the following operation on each element of the array at most once:

Add an integer in the range [-k, k] to the element.

Return the maximum possible number of distinct elements in nums after performing the operations.

Example 1:

Input: nums = [1,2,2,3,3,4], k = 2
Output: 6
Explanation:
nums changes to [-1, 0, 1, 2, 3, 4] after performing operations on the first four elements.

Example 2:

Input: nums = [4,4,4,4], k = 1
Output: 3
Explanation:
By adding -1 to nums[0] and 1 to nums[1], nums changes to [3, 5, 4, 4].

Constraints:

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

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

SOLUTION:

Question:
PROBLEM:
You are given an integer array nums and an integer k.
You are allowed to perform the following operation on each element of the array at most once:

Add an integer in the range [-k, k] to the element.

Return the maximum possible number of distinct elements in nums after performing the operations.

Example 1:

Input: nums = [1,2,2,3,3,4], k = 2
Output: 6
Explanation:
nums changes to [-1, 0, 1, 2, 3, 4] after performing operations on the first four elements.

Example 2:

Input: nums = [4,4,4,4], k = 1
Output: 3
Explanation:
By adding -1 to nums[0] and 1 to nums[1], nums changes to [3, 5, 4, 4].

Constraints:

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

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

SOLUTION:

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

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

assert Solution().maxDistinctElements(nums = [1,1,1,1,1], k = 0) == 1

assert Solution().maxDistinctElements(nums = [4,4,4,4], k = 1) == 3

assert Solution().maxDistinctElements(nums = [1,1,1,1,1], k = 2) == 5

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

assert Solution().maxDistinctElements(nums = [5,5,5,5,5,5,5,5,5,5], k = 2) == 5

assert Solution().maxDistinctElements(nums = [1,2,3,4,5], k = 0) == 5

assert Solution().maxDistinctElements(nums = [1,1,1,1,1,1,1,1,1,1], k = 1) == 3

assert Solution().maxDistinctElements(nums = [10,10,20,20,30], k = 5) == 5

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

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

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

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

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

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

assert Solution().maxDistinctElements(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 0) == 1

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

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

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

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

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

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

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

assert Solution().maxDistinctElements(nums = [1, 3, 3, 5, 7, 8, 8, 10, 12, 12], k = 2) == 10

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

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

assert Solution().maxDistinctElements(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 2) == 5

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

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

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

assert Solution().maxDistinctElements(nums = [1, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 2) == 6

Answer: assert Solution().maxDistinctElements(nums = [1,1000000000,1,1000000000], k = 1000000000) == 4

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

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

Answer: assert Solution().maxDistinctElements(nums = [1,1,1,1,1], k = 0) == 1

Answer: assert Solution().maxDistinctElements(nums = [4,4,4,4], k = 1) == 3

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

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

Answer: assert Solution().maxDistinctElements(nums = [5,5,5,5,5,5,5,5,5,5], k = 2) == 5

Answer: assert Solution().maxDistinctElements(nums = [1,2,3,4,5], k = 0) == 5

Answer: assert Solution().maxDistinctElements(nums = [1,1,1,1,1,1,1,1,1,1], k = 1) == 3

Answer: assert Solution().maxDistinctElements(nums = [10,10,20,20,30], k = 5) == 5

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

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

Answer: assert Solution().maxDistinctElements(nums = [999999999,999999999,999999999,999999999,999999999], k = 0) == 1

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 3) == 16

Answer: assert Solution().maxDistinctElements(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) == 25

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

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

Answer: assert Solution().maxDistinctElements(nums = [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57, 61, 65, 69, 73, 77], k = 2) == 20

Answer: assert Solution().maxDistinctElements(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39], k = 10) == 20

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

Answer: assert Solution().maxDistinctElements(nums = [1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000], k = 500000000) == 10

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

Answer: assert Solution().maxDistinctElements(nums = [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39], k = 10) == 20

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

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

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 1) == 12

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1) == 3

Answer: assert Solution().maxDistinctElements(nums = [1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9,10,10], k = 3) == 16

Answer: assert Solution().maxDistinctElements(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 2) == 5

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

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

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

Answer: assert Solution().maxDistinctElements(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 1000000000) == 20

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

Answer: assert Solution().maxDistinctElements(nums = [5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9], k = 2) == 9

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

Answer: assert Solution().maxDistinctElements(nums = [10, 20, 20, 20, 30, 30, 40, 50, 60, 70, 80, 90, 100], k = 15) == 13

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

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

Answer: assert Solution().maxDistinctElements(nums = [100, 100, 100, 100, 100, 100, 100, 100, 100, 100], k = 50) == 10

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

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

Answer: assert Solution().maxDistinctElements(nums = [1,2,2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,6,6], k = 0) == 6

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

Answer: assert Solution().maxDistinctElements(nums = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 3) == 7

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

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

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

Answer: assert Solution().maxDistinctElements(nums = [1000000000, 1000000000, 1000000000, 1000000000], k = 1) == 3

Answer: assert Solution().maxDistinctElements(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], k = 1000000000) == 10

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

Answer: assert Solution().maxDistinctElements(nums = [1000000000, 1000000000, 1000000000, 1000000000, 1000000000], k = 0) == 1

Answer: assert Solution().maxDistinctElements(nums = [1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10], k = 9) == 20

Answer: assert Solution().maxDistinctElements(nums = [10, 10, 10, 11, 11, 12, 13, 13, 14, 14], k = 3) == 10

Answer: assert Solution().maxDistinctElements(nums = [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 = 1) == 12

Answer: assert Solution().maxDistinctElements(nums = [1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11], k = 3) == 17

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

Answer: assert Solution().maxDistinctElements(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], k = 5) == 30

Answer: assert Solution().maxDistinctElements(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], k = 150) == 10

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 2) == 14

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

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

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

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

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

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

Answer: assert Solution().maxDistinctElements(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 = 1) == 25

Answer: assert Solution().maxDistinctElements(nums = [100, 200, 200, 300, 300, 400, 500, 500, 600, 700, 700, 800, 900, 1000], k = 100) == 14

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

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

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

Answer: assert Solution().maxDistinctElements(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, 15, 15], k = 2) == 19

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

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

Answer: assert Solution().maxDistinctElements(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], k = 5) == 11

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

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1000000000) == 20

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

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

Answer: assert Solution().maxDistinctElements(nums = [1,3,3,3,3,5,5,5,7,7,9,9,11,11,13,13,15,15,17,17], k = 2) == 20

Answer: assert Solution().maxDistinctElements(nums = [10, 10, 20, 20, 20, 30, 30, 30, 30, 40, 40, 40, 40, 40, 50, 50, 50, 50, 50, 50, 60, 60, 60, 60, 60, 60, 60, 70, 70, 70, 70, 70, 70, 70, 70], k = 5) == 35

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

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 5) == 11

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

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

Answer: assert Solution().maxDistinctElements(nums = [5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8], k = 3) == 10

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

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

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1) == 3

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

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

Answer: assert Solution().maxDistinctElements(nums = [1, 3, 3, 5, 7, 8, 8, 10, 12, 12], k = 2) == 10

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

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 0) == 1

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

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

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

Answer: assert Solution().maxDistinctElements(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 2) == 5

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

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

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

Answer: assert Solution().maxDistinctElements(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], k = 25) == 50

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

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

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

Answer: assert Solution().maxDistinctElements(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 = 1) == 7

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

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

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

Answer: assert Solution().maxDistinctElements(nums = [1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 10, 11, 12, 13, 13, 14], k = 1) == 16

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 10) == 21

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

Answer: assert Solution().maxDistinctElements(nums = [1000000, 1000001, 1000001, 1000002, 1000003, 1000003, 1000004, 1000005], k = 1000000) == 8

Answer: assert Solution().maxDistinctElements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 100) == 30

Answer: assert Solution().maxDistinctElements(nums = [1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 1) == 12

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

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

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

Answer: assert Solution().maxDistinctElements(nums = [5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10], k = 2) == 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 maxDistinctElements(self, nums: List[int], k: int) -> int:
        nums.sort()
        ans = 0
        pre = -inf
        for x in nums:
            cur = min(x + k, max(x - k, pre + 1))
            if cur > pre:
                ans += 1
                pre = cur
        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 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 <= 1000
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 <= 1000
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 = "111000", numOps = 1) == 3

assert Solution().minLength(s = "11111", numOps = 2) == 1

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

assert Solution().minLength(s = "1100110011", numOps = 3) == 2

assert Solution().minLength(s = "001100", numOps = 2) == 2

assert Solution().minLength(s = "111000111", numOps = 3) == 1

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

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

assert Solution().minLength(s = "11001100", numOps = 4) == 1

assert Solution().minLength(s = "1111", numOps = 1) == 2

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

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

assert Solution().minLength(s = "101010", numOps = 5) == 1

assert Solution().minLength(s = "0000000", numOps = 0) == 7

assert Solution().minLength(s = "111100001111", numOps = 6) == 1

assert Solution().minLength(s = "000111", numOps = 2) == 1

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

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

assert Solution().minLength(s = "110011", numOps = 4) == 1

assert Solution().minLength(s = "001100", numOps = 1) == 2

assert Solution().minLength(s = "1111111111", numOps = 5) == 1

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

assert Solution().minLength(s = "0101010101", numOps = 3) == 1

assert Solution().minLength(s = "0011001100", numOps = 6) == 1

assert Solution().minLength(s = "0000000000", numOps = 5) == 1

assert Solution().minLength(s = "1010101010", numOps = 3) == 1

assert Solution().minLength(s = "1110010010100101", numOps = 5) == 2

assert Solution().minLength(s = "1111000011", numOps = 5) == 1

assert Solution().minLength(s = "111110000011111", numOps = 5) == 2

assert Solution().minLength(s = "1010101010101010", numOps = 5) == 1

assert Solution().minLength(s = "1111000000", numOps = 4) == 2

assert Solution().minLength(s = "1001001001001001", numOps = 7) == 2

assert Solution().minLength(s = "0110110110110110", numOps = 7) == 2

Answer: assert Solution().minLength(s = "111000", numOps = 1) == 3

Answer: assert Solution().minLength(s = "11111", numOps = 2) == 1

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

Answer: assert Solution().minLength(s = "1100110011", numOps = 3) == 2

Answer: assert Solution().minLength(s = "001100", numOps = 2) == 2

Answer: assert Solution().minLength(s = "111000111", numOps = 3) == 1

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

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

Answer: assert Solution().minLength(s = "11001100", numOps = 4) == 1

Answer: assert Solution().minLength(s = "1111", numOps = 1) == 2

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

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

Answer: assert Solution().minLength(s = "101010", numOps = 5) == 1

Answer: assert Solution().minLength(s = "0000000", numOps = 0) == 7

Answer: assert Solution().minLength(s = "111100001111", numOps = 6) == 1

Answer: assert Solution().minLength(s = "000111", numOps = 2) == 1

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

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

Answer: assert Solution().minLength(s = "110011", numOps = 4) == 1

Answer: assert Solution().minLength(s = "001100", numOps = 1) == 2

Answer: assert Solution().minLength(s = "10101010101010101010", numOps = 3) == 1

Answer: assert Solution().minLength(s = "111111111111111111111111", numOps = 7) == 3

Answer: assert Solution().minLength(s = "0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", numOps = 100) == 1

Answer: assert Solution().minLength(s = "111111111100000000001111111111", numOps = 10) == 2

Answer: assert Solution().minLength(s = "10101010101010101010101010101010", numOps = 15) == 1

Answer: assert Solution().minLength(s = "10101010101010101010101010101010101010101010", numOps = 25) == 1

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

Answer: assert Solution().minLength(s = "1111111111", numOps = 5) == 1

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

Answer: assert Solution().minLength(s = "1111000011110000111100001111000011110000111100001111000011110000111100001111", numOps = 30) == 2

Answer: assert Solution().minLength(s = "0000111100001111000011110000", numOps = 20) == 1

Answer: assert Solution().minLength(s = "01010101010101010101", numOps = 5) == 1

Answer: assert Solution().minLength(s = "11001100110011001100", numOps = 20) == 1

Answer: assert Solution().minLength(s = "000000000000000000000000", numOps = 10) == 2

Answer: assert Solution().minLength(s = "1110000011100000111", numOps = 8) == 1

Answer: assert Solution().minLength(s = "010010010010010010010010", numOps = 7) == 2

Answer: assert Solution().minLength(s = "111111000000111111000000111111000000111111000000", numOps = 15) == 3

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

Answer: assert Solution().minLength(s = "10010010010010010010", numOps = 12) == 1

Answer: assert Solution().minLength(s = "00001111000011110000", numOps = 8) == 2

Answer: assert Solution().minLength(s = "1111111111111111111111111111111111111111111111111111111111111111111111111111", numOps = 75) == 1

Answer: assert Solution().minLength(s = "000111000111000111000111000111000111000111000111", numOps = 15) == 3

Answer: assert Solution().minLength(s = "11111000001111100000", numOps = 5) == 2

Answer: assert Solution().minLength(s = "111110000011111000001111100000", numOps = 18) == 1

Answer: assert Solution().minLength(s = "01010101010101010101", numOps = 3) == 1

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

Answer: assert Solution().minLength(s = "11100011100011100011", numOps = 8) == 1

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

Answer: assert Solution().minLength(s = "01010101010101010101010101010101010101010101010101010101010101010101", numOps = 30) == 1

Answer: assert Solution().minLength(s = "11111111111111111111", numOps = 8) == 2

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

Answer: assert Solution().minLength(s = "100100100100100100100100", numOps = 6) == 2

Answer: assert Solution().minLength(s = "11111000000001111111", numOps = 7) == 2

Answer: assert Solution().minLength(s = "100100100100100100100100100100100100100100100100100100100100100100100100100", numOps = 25) == 2

Answer: assert Solution().minLength(s = "000000000000000000000000", numOps = 30) == 1

Answer: assert Solution().minLength(s = "0101010101", numOps = 3) == 1

Answer: assert Solution().minLength(s = "01010101010101010101", numOps = 0) == 1

Answer: assert Solution().minLength(s = "00110011001100110011", numOps = 15) == 1

Answer: assert Solution().minLength(s = "0101010101010101010101010101010101010101", numOps = 20) == 1

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

Answer: assert Solution().minLength(s = "10101010101010101010", numOps = 7) == 1

Answer: assert Solution().minLength(s = "00001111000011110000", numOps = 5) == 2

Answer: assert Solution().minLength(s = "11111111111111111111", numOps = 15) == 1

Answer: assert Solution().minLength(s = "0011001100", numOps = 6) == 1

Answer: assert Solution().minLength(s = "010101010101010101010101010101", numOps = 15) == 1

Answer: assert Solution().minLength(s = "111111111111111111111111", numOps = 30) == 1

Answer: assert Solution().minLength(s = "00000000000000000000", numOps = 10) == 1

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

Answer: assert Solution().minLength(s = "00000000000000000000", numOps = 7) == 2

Answer: assert Solution().minLength(s = "111001100110011001100110011", numOps = 12) == 2

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

Answer: assert Solution().minLength(s = "11110000111100001111", numOps = 10) == 1

Answer: assert Solution().minLength(s = "11100011100011100011", numOps = 9) == 1

Answer: assert Solution().minLength(s = "010101010101010101010101", numOps = 0) == 1

Answer: assert Solution().minLength(s = "1010101010101010101010", numOps = 8) == 1

Answer: assert Solution().minLength(s = "0000000000", numOps = 5) == 1

Answer: assert Solution().minLength(s = "101010101010101010101010101010", numOps = 15) == 1

Answer: assert Solution().minLength(s = "1010101010", numOps = 3) == 1

Answer: assert Solution().minLength(s = "11001100110011001100", numOps = 8) == 2

Answer: assert Solution().minLength(s = "11111111110000000000", numOps = 5) == 3

Answer: assert Solution().minLength(s = "010101010101010101010101", numOps = 10) == 1

Answer: assert Solution().minLength(s = "00000000001111111111000000000011111111", numOps = 15) == 2

Answer: assert Solution().minLength(s = "1111100000111110000011111", numOps = 10) == 1

Answer: assert Solution().minLength(s = "00001111000011110000", numOps = 10) == 1

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

Answer: assert Solution().minLength(s = "00011100011100011100", numOps = 9) == 1

Answer: assert Solution().minLength(s = "11110101010101010101010101010101010101010101", numOps = 12) == 1

Answer: assert Solution().minLength(s = "111111111111111111111111111111111111111111111111", numOps = 45) == 1

Answer: assert Solution().minLength(s = "01010101010101010101010101010101010101010101", numOps = 20) == 1

Answer: assert Solution().minLength(s = "01010101010101010101010101010101010101010101010101010101010101010101010101", numOps = 30) == 1

Answer: assert Solution().minLength(s = "000000000011111111110000000000", numOps = 10) == 2

Answer: assert Solution().minLength(s = "10101010101010101010", numOps = 8) == 1

Answer: assert Solution().minLength(s = "11100011100011100011", numOps = 12) == 1

Answer: assert Solution().minLength(s = "11011011011011011011", numOps = 10) == 1

Answer: assert Solution().minLength(s = "0000011111000001111100000", numOps = 10) == 1

Answer: assert Solution().minLength(s = "1110010010100101", numOps = 5) == 2

Answer: assert Solution().minLength(s = "000000000000000000", numOps = 10) == 1

Answer: assert Solution().minLength(s = "1111000011110000111100001111", numOps = 12) == 2

Answer: assert Solution().minLength(s = "001100110011001100110011", numOps = 10) == 2

Answer: assert Solution().minLength(s = "110011001100110011001100", numOps = 5) == 2

Answer: assert Solution().minLength(s = "1111000011", numOps = 5) == 1

Answer: assert Solution().minLength(s = "10101010101010101010101010101010", numOps = 18) == 1

Answer: assert Solution().minLength(s = "111110000011111", numOps = 5) == 2

Answer: assert Solution().minLength(s = "1010101010101010", numOps = 5) == 1

Answer: assert Solution().minLength(s = "1111000000", numOps = 4) == 2

Answer: assert Solution().minLength(s = "101010101010101010101010", numOps = 5) == 1

Answer: assert Solution().minLength(s = "1110000001110000011100000111000001110000011100000111000001110000011100000111", numOps = 20) == 2

Answer: assert Solution().minLength(s = "00001111000011110000111100001111000011110000", numOps = 20) == 2

Answer: assert Solution().minLength(s = "1101010101010101", numOps = 10) == 1

Answer: assert Solution().minLength(s = "1001001001001001", numOps = 7) == 2

Answer: assert Solution().minLength(s = "0110110110110110", numOps = 7) == 2

Answer: assert Solution().minLength(s = "00000000000000000000", numOps = 15) == 1

Answer: assert Solution().minLength(s = "000000000011111111110000000000", numOps = 8) == 3

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

Answer: assert Solution().minLength(s = "010101010101010101010101", numOps = 15) == 1

Answer: assert Solution().minLength(s = "01010101010101010101", numOps = 10) == 1

Answer: assert Solution().minLength(s = "00000000000000000000", numOps = 20) == 1

Answer: assert Solution().minLength(s = "11110000111100001111", numOps = 12) == 1

Answer: assert Solution().minLength(s = "00001111111100001111", numOps = 6) == 2

Answer: assert Solution().minLength(s = "100000000000000000000000000000000000000000000000000", numOps = 50) == 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)