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:
There is an infinite 2D plane.
You are given a positive integer k. You are also given a 2D array queries, which contains the following queries:

queries[i] = [x, y]: Build an obstacle at coordinate (x, y) in the plane. It is guaranteed that there is no obstacle at this coordinate when this query is made.

After each query, you need to find the distance of the kth nearest obstacle from the origin.
Return an integer array results where results[i] denotes the kth nearest obstacle after query i, or results[i] == -1 if there are less than k obstacles.
Note that initially there are no obstacles anywhere.
The distance of an obstacle at coordinate (x, y) from the origin is given by |x| + |y|.

Example 1:

Input: queries = [[1,2],[3,4],[2,3],[-3,0]], k = 2
Output: [-1,7,5,3]
Explanation:

Initially, there are 0 obstacles.
After queries[0], there are less than 2 obstacles.
After queries[1], there are obstacles at distances 3 and 7.
After queries[2], there are obstacles at distances 3, 5, and 7.
After queries[3], there are obstacles at distances 3, 3, 5, and 7.

Example 2:

Input: queries = [[5,5],[4,4],[3,3]], k = 1
Output: [10,8,6]
Explanation:

After queries[0], there is an obstacle at distance 10.
After queries[1], there are obstacles at distances 8 and 10.
After queries[2], there are obstacles at distances 6, 8, and 10.

Constraints:

1 <= queries.length <= 2 * 105
All queries[i] are unique.
-109 <= queries[i][0], queries[i][1] <= 109
1 <= k <= 105

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

SOLUTION:

Question:
PROBLEM:
There is an infinite 2D plane.
You are given a positive integer k. You are also given a 2D array queries, which contains the following queries:

queries[i] = [x, y]: Build an obstacle at coordinate (x, y) in the plane. It is guaranteed that there is no obstacle at this coordinate when this query is made.

After each query, you need to find the distance of the kth nearest obstacle from the origin.
Return an integer array results where results[i] denotes the kth nearest obstacle after query i, or results[i] == -1 if there are less than k obstacles.
Note that initially there are no obstacles anywhere.
The distance of an obstacle at coordinate (x, y) from the origin is given by |x| + |y|.

Example 1:

Input: queries = [[1,2],[3,4],[2,3],[-3,0]], k = 2
Output: [-1,7,5,3]
Explanation:

Initially, there are 0 obstacles.
After queries[0], there are less than 2 obstacles.
After queries[1], there are obstacles at distances 3 and 7.
After queries[2], there are obstacles at distances 3, 5, and 7.
After queries[3], there are obstacles at distances 3, 3, 5, and 7.

Example 2:

Input: queries = [[5,5],[4,4],[3,3]], k = 1
Output: [10,8,6]
Explanation:

After queries[0], there is an obstacle at distance 10.
After queries[1], there are obstacles at distances 8 and 10.
After queries[2], there are obstacles at distances 6, 8, and 10.

Constraints:

1 <= queries.length <= 2 * 105
All queries[i] are unique.
-109 <= queries[i][0], queries[i][1] <= 109
1 <= k <= 105

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

SOLUTION:

Answer: assert Solution().resultsArray(queries = [[5,5],[4,4],[3,3]], k = 1) == [10, 8, 6]

Answer: assert Solution().resultsArray(queries = [[-1000000000,1000000000],[1000000000,-1000000000]], k = 1) == [2000000000, 2000000000]

Answer: assert Solution().resultsArray(queries = [[-1,1],[0,0],[1,-1]], k = 3) == [-1, -1, 2]

Answer: assert Solution().resultsArray(queries = [[10,10],[20,20],[30,30],[40,40]], k = 4) == [-1, -1, -1, 80]

Answer: assert Solution().resultsArray(queries = [[1,2],[3,4],[2,3],[-3,0]], k = 2) == [-1, 7, 5, 3]

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

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

Answer: assert Solution().resultsArray(queries = [[-1,0],[0,1],[1,0],[0,-1],[-2,0],[0,-2],[2,0],[0,2],[-3,0],[0,-3],[3,0],[0,3]], k = 4) == [-1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[1000000000,1000000000],[1000000000,-1000000000],[-1000000000,1000000000],[-1000000000,-1000000000]], k = 1) == [2000000000, 2000000000, 2000000000, 2000000000]

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

Answer: assert Solution().resultsArray(queries = [[-100,-100],[100,-100],[-100,100],[100,100]], k = 2) == [-1, 200, 200, 200]

Answer: assert Solution().resultsArray(queries = [[100, 100], [200, 200], [300, 300], [400, 400], [500, 500], [600, 600], [700, 700]], k = 5) == [-1, -1, -1, -1, 1000, 1000, 1000]

Answer: assert Solution().resultsArray(queries = [[-2,-3],[-4,-5],[6,7],[8,9],[0,0]], k = 3) == [-1, -1, 13, 13, 9]

Answer: assert Solution().resultsArray(queries = [[1,1],[1,2],[2,1],[2,2],[1,3],[3,1],[2,3],[3,2],[3,3]], k = 3) == [-1, -1, 3, 3, 3, 3, 3, 3, 3]

Answer: assert Solution().resultsArray(queries = [[0,1],[1,0],[0,-1],[-1,0],[2,1],[1,2],[1,-2],[-2,1],[-1,-2],[2,-1],[2,0],[0,2],[-2,0],[0,-2],[3,0],[0,3],[-3,0],[0,-3]], k = 7) == [-1, -1, -1, -1, -1, -1, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2]

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

Answer: assert Solution().resultsArray(queries = [[0,1],[1,0],[1,1],[-1,-1],[-1,0],[0,-1]], k = 3) == [-1, -1, 2, 2, 1, 1]

Answer: assert Solution().resultsArray(queries = [[-5,5],[5,-5],[0,10],[10,0],[-10,-10]], k = 3) == [-1, -1, 10, 10, 10]

Answer: assert Solution().resultsArray(queries = [[1000000000,1000000000],[-1000000000,-1000000000],[500000000,500000000],[-500000000,-500000000]], k = 2) == [-1, 2000000000, 2000000000, 1000000000]

Answer: assert Solution().resultsArray(queries = [[-50, -50], [-50, 50], [50, -50], [50, 50], [0, 0], [0, 100], [0, -100], [100, 0], [-100, 0]], k = 2) == [-1, 100, 100, 100, 100, 100, 100, 100, 100]

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

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

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

Answer: assert Solution().resultsArray(queries = [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10], [11, 12], [13, 14], [15, 16], [17, 18], [19, 20]], k = 10) == [-1, -1, -1, -1, -1, -1, -1, -1, -1, 39]

Answer: assert Solution().resultsArray(queries = [[1,0],[0,1],[-1,0],[0,-1],[2,2],[2,-2],[-2,2],[-2,-2],[3,3],[3,-3],[-3,3],[-3,-3]], k = 4) == [-1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[999999999,999999999],[999999998,999999998],[999999997,999999997],[999999996,999999996],[999999995,999999995],[999999994,999999994]], k = 5) == [-1, -1, -1, -1, 1999999998, 1999999996]

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

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

Answer: assert Solution().resultsArray(queries = [[100,100],[200,200],[300,300],[400,400],[500,500]], k = 1) == [200, 200, 200, 200, 200]

Answer: assert Solution().resultsArray(queries = [[0,0],[1,0],[0,1],[-1,0],[0,-1],[2,0],[0,2],[-2,0],[0,-2],[3,0],[0,3],[-3,0],[0,-3]], k = 1) == [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

Answer: assert Solution().resultsArray(queries = [[1,2],[2,1],[3,3],[4,4],[5,5],[6,6],[7,7],[8,8],[9,9],[10,10]], k = 10) == [-1, -1, -1, -1, -1, -1, -1, -1, -1, 20]

Answer: assert Solution().resultsArray(queries = [[1000000000,0],[0,1000000000],[-1000000000,0],[0,-1000000000]], k = 2) == [-1, 1000000000, 1000000000, 1000000000]

Answer: assert Solution().resultsArray(queries = [[-1,-1],[1,-1],[-1,1],[1,1],[-2,-2],[2,-2],[-2,2],[2,2],[-3,-3],[3,-3],[-3,3],[3,3]], k = 5) == [-1, -1, -1, -1, 4, 4, 4, 4, 4, 4, 4, 4]

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

Answer: assert Solution().resultsArray(queries = [[0, 0], [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]], k = 7) == [-1, -1, -1, -1, -1, -1, 12, 12, 12, 12, 12, 12, 12, 12, 12]

Answer: assert Solution().resultsArray(queries = [[-1000,-2000],[-2000,-3000],[-3000,-4000],[-4000,-5000],[-5000,-6000],[-6000,-7000],[-7000,-8000],[-8000,-9000]], k = 3) == [-1, -1, 7000, 7000, 7000, 7000, 7000, 7000]

Answer: assert Solution().resultsArray(queries = [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10], [11, 12], [13, 14], [15, 16], [17, 18], [19, 20], [21, 22]], k = 11) == [-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 43]

Answer: assert Solution().resultsArray(queries = [[0,0],[1,0],[0,1],[-1,0],[0,-1],[2,0],[0,2],[-2,0],[0,-2],[3,0],[0,3]], k = 5) == [-1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[0,0],[10,10],[-10,-10],[10,-10],[-10,10]], k = 5) == [-1, -1, -1, -1, 20]

Answer: assert Solution().resultsArray(queries = [[10,10],[20,20],[30,30],[40,40],[50,50],[60,60],[70,70],[80,80],[90,90],[100,100]], k = 5) == [-1, -1, -1, -1, 100, 100, 100, 100, 100, 100]

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

Answer: assert Solution().resultsArray(queries = [[-1,0],[0,1],[1,0],[0,-1],[-2,0],[0,-2],[2,0],[0,2],[-3,0],[0,-3],[3,0],[0,3]], k = 6) == [-1, -1, -1, -1, -1, 2, 2, 2, 2, 2, 2, 2]

Answer: assert Solution().resultsArray(queries = [[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]], k = 7) == [-1, -1, -1, -1, -1, -1, 14, 14, 14, 14, 14, 14, 14, 14, 14]

Answer: assert Solution().resultsArray(queries = [[-5,5],[-4,4],[-3,3],[-2,2],[-1,1],[0,0],[1,1],[2,2],[3,3],[4,4],[5,5]], k = 5) == [-1, -1, -1, -1, 10, 8, 6, 4, 4, 4, 4]

Answer: assert Solution().resultsArray(queries = [[-1,0],[0,-1],[1,0],[0,1],[-2,0],[0,-2],[2,0],[0,2],[-3,0],[0,-3],[3,0],[0,3]], k = 4) == [-1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[10,20],[20,30],[30,40],[40,50],[50,60],[60,70],[70,80],[80,90],[90,100]], k = 7) == [-1, -1, -1, -1, -1, -1, 150, 150, 150]

Answer: assert Solution().resultsArray(queries = [[1,2],[2,1],[1,-2],[-2,1],[3,-3],[-3,3],[2,2],[-2,-2],[1,1],[-1,-1]], k = 5) == [-1, -1, -1, -1, 6, 6, 4, 4, 3, 3]

Answer: assert Solution().resultsArray(queries = [[1,1],[-1,-1],[1,-1],[-1,1],[2,2],[-2,-2],[2,-2],[-2,2],[3,3],[-3,-3],[3,-3],[-3,3]], k = 6) == [-1, -1, -1, -1, -1, 4, 4, 4, 4, 4, 4, 4]

Answer: assert Solution().resultsArray(queries = [[-10,-10],[-20,-20],[-30,-30],[-40,-40],[-50,-50],[-60,-60],[-70,-70],[-80,-80],[-90,-90],[-100,-100]], k = 8) == [-1, -1, -1, -1, -1, -1, -1, 160, 160, 160]

Answer: assert Solution().resultsArray(queries = [[1,0],[0,1],[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]], k = 7) == [-1, -1, -1, -1, -1, -1, 10, 10, 10, 10, 10, 10, 10, 10, 10]

Answer: assert Solution().resultsArray(queries = [[-1000000000, 0], [1000000000, 0], [0, -1000000000], [0, 1000000000]], k = 3) == [-1, -1, 1000000000, 1000000000]

Answer: assert Solution().resultsArray(queries = [[0,0],[1,0],[0,1],[1,1],[-1,0],[0,-1],[-1,-1],[-1,1],[1,-1],[0,2]], k = 5) == [-1, -1, -1, -1, 2, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[0,0],[10,10],[20,20],[30,30],[40,40],[50,50],[60,60],[70,70],[80,80],[90,90],[100,100]], k = 6) == [-1, -1, -1, -1, -1, 100, 100, 100, 100, 100, 100]

Answer: assert Solution().resultsArray(queries = [[100,200],[200,300],[300,400],[400,500],[500,600],[600,700],[700,800],[800,900]], k = 7) == [-1, -1, -1, -1, -1, -1, 1500, 1500]

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

Answer: assert Solution().resultsArray(queries = [[1,1],[-1,-1],[1,-1],[-1,1],[0,0],[2,2],[-2,-2],[2,-2],[-2,2]], k = 4) == [-1, -1, -1, 2, 2, 2, 2, 2, 2]

Answer: assert Solution().resultsArray(queries = [[0,0],[0,0],[0,0],[0,0],[0,0]], k = 5) == [-1, -1, -1, -1, 0]

Answer: assert Solution().resultsArray(queries = [[1000000000,0],[0,1000000000],[-1000000000,0],[0,-1000000000],[500000000,500000000],[-500000000,-500000000]], k = 2) == [-1, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000]

Answer: assert Solution().resultsArray(queries = [[10,10],[20,20],[30,30],[40,40],[50,50],[60,60],[70,70],[80,80],[90,90],[100,100]], k = 10) == [-1, -1, -1, -1, -1, -1, -1, -1, -1, 200]

Answer: assert Solution().resultsArray(queries = [[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]], k = 5) == [-1, -1, -1, -1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10]

Answer: assert Solution().resultsArray(queries = [[-1,0],[0,-1],[1,0],[0,1],[0,0],[2,0],[0,2],[-2,0],[0,-2]], k = 3) == [-1, -1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[-10,-10],[-20,-20],[-30,-30],[-40,-40],[-50,-50],[-60,-60],[-70,-70],[-80,-80],[-90,-90],[-100,-100]], k = 4) == [-1, -1, -1, 80, 80, 80, 80, 80, 80, 80]

Answer: assert Solution().resultsArray(queries = [[1000000,1000000],[2000000,2000000],[3000000,3000000],[4000000,4000000],[5000000,5000000],[6000000,6000000],[7000000,7000000],[8000000,8000000]], k = 4) == [-1, -1, -1, 8000000, 8000000, 8000000, 8000000, 8000000]

Answer: assert Solution().resultsArray(queries = [[1000000000,-1000000000],[-1000000000,1000000000],[0,0],[1,1],[-1,-1],[2,2],[-2,-2]], k = 3) == [-1, -1, 2000000000, 2000000000, 2, 2, 2]

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

Answer: assert Solution().resultsArray(queries = [[-5,4],[-4,-5],[4,5],[5,-4],[2,3],[3,2],[-2,-3],[-3,-2]], k = 2) == [-1, 9, 9, 9, 9, 5, 5, 5]

Answer: assert Solution().resultsArray(queries = [[1,0],[0,1],[-1,0],[0,-1],[1,1],[-1,-1],[-1,1],[1,-1]], k = 2) == [-1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[1000000000,-1000000000],[-1000000000,1000000000],[0,0]], k = 2) == [-1, 2000000000, 2000000000]

Answer: assert Solution().resultsArray(queries = [[100,200],[200,300],[300,400],[400,500],[500,600],[600,700],[700,800],[800,900]], k = 4) == [-1, -1, -1, 900, 900, 900, 900, 900]

Answer: assert Solution().resultsArray(queries = [[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]], k = 10) == [-1, -1, -1, -1, -1, -1, -1, -1, -1, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20, 20]

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

Answer: assert Solution().resultsArray(queries = [[1000000000,-1000000000],[-1000000000,1000000000],[0,0],[0,1],[1,0]], k = 3) == [-1, -1, 2000000000, 2000000000, 1]

Answer: assert Solution().resultsArray(queries = [[1,1],[-1,-1],[1,-1],[-1,1],[2,2],[-2,-2],[2,-2],[-2,2],[3,3],[-3,-3]], k = 6) == [-1, -1, -1, -1, -1, 4, 4, 4, 4, 4]

Answer: assert Solution().resultsArray(queries = [[0,0],[1,0],[0,1],[-1,0],[0,-1],[2,0],[0,2],[-2,0],[0,-2],[3,0],[0,3],[-3,0],[0,-3]], k = 4) == [-1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[1000000000,1000000000],[-1000000000,-1000000000],[500000000,500000000]], k = 2) == [-1, 2000000000, 2000000000]

Answer: assert Solution().resultsArray(queries = [[-1,0],[0,-1],[1,0],[0,1],[-1,-1],[1,1],[0,2],[-2,0],[2,0],[0,-2]], k = 4) == [-1, -1, -1, 1, 1, 1, 1, 1, 1, 1]

Answer: assert Solution().resultsArray(queries = [[-5,0],[0,5],[5,0],[0,-5],[1,1],[-1,-1],[1,-1],[-1,1]], k = 3) == [-1, -1, 5, 5, 5, 5, 2, 2]

Answer: assert Solution().resultsArray(queries = [[0, 1], [0, 2], [0, 3], [0, 4], [0, 5], [0, 6], [0, 7], [0, 8], [0, 9], [0, 10], [0, 11], [0, 12]], k = 6) == [-1, -1, -1, -1, -1, 6, 6, 6, 6, 6, 6, 6]

Answer: assert Solution().resultsArray(queries = [[1,0],[0,1],[-1,0],[0,-1],[1,1],[-1,-1],[1,-1],[-1,1],[2,2],[-2,-2]], k = 5) == [-1, -1, -1, -1, 2, 2, 2, 2, 2, 2]

Answer: assert Solution().resultsArray(queries = [[100,1],[1,100],[10,10],[1000,1000],[10000,10000],[100000,100000],[1000000,1000000],[10000000,10000000],[100000000,100000000],[1000000000,1000000000]], k = 7) == [-1, -1, -1, -1, -1, -1, 2000000, 2000000, 2000000, 2000000]

Answer: assert Solution().resultsArray(queries = [[1,2],[2,1],[-1,-2],[-2,-1],[1,-2],[2,-1],[-1,2],[-2,1]], k = 3) == [-1, -1, 3, 3, 3, 3, 3, 3]

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

Answer: assert Solution().resultsArray(queries = [[1000000000,1000000000],[500000000,-500000000],[-1000000000,1000000000],[0,0],[1,1],[2,2]], k = 3) == [-1, -1, 2000000000, 2000000000, 1000000000, 4]

Answer: assert Solution().resultsArray(queries = [[-5,5],[-4,4],[-3,3],[-2,2],[-1,1],[0,0],[1,1],[2,2],[3,3],[4,4],[5,5]], k = 3) == [-1, -1, 10, 8, 6, 4, 2, 2, 2, 2, 2]

Answer: assert Solution().resultsArray(queries = [[10,10],[20,20],[30,30],[40,40],[50,50],[60,60],[70,70],[80,80],[90,90],[100,100]], k = 7) == [-1, -1, -1, -1, -1, -1, 140, 140, 140, 140]

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 resultsArray(self, queries: List[List[int]], k: int) -> List[int]:
        ans = []
        pq = []
        for i, (x, y) in enumerate(queries):
            heappush(pq, -(abs(x) + abs(y)))
            if i >= k:
                heappop(pq)
            ans.append(-pq[0] if i >= k - 1 else -1)
        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 2D matrix grid consisting of positive integers.
You have to select one or more cells from the matrix such that the following conditions are satisfied:

No two selected cells are in the same row of the matrix.
The values in the set of selected cells are unique.

Your score will be the sum of the values of the selected cells.
Return the maximum score you can achieve.

Example 1:

Input: grid = [[1,2,3],[4,3,2],[1,1,1]]
Output: 8
Explanation:

We can select the cells with values 1, 3, and 4 that are colored above.

Example 2:

Input: grid = [[8,7,6],[8,3,2]]
Output: 15
Explanation:

We can select the cells with values 7 and 8 that are colored above.

Constraints:

1 <= grid.length, grid[i].length <= 10
1 <= grid[i][j] <= 100

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

SOLUTION:

Question:
PROBLEM:
You are given a 2D matrix grid consisting of positive integers.
You have to select one or more cells from the matrix such that the following conditions are satisfied:

No two selected cells are in the same row of the matrix.
The values in the set of selected cells are unique.

Your score will be the sum of the values of the selected cells.
Return the maximum score you can achieve.

Example 1:

Input: grid = [[1,2,3],[4,3,2],[1,1,1]]
Output: 8
Explanation:

We can select the cells with values 1, 3, and 4 that are colored above.

Example 2:

Input: grid = [[8,7,6],[8,3,2]]
Output: 15
Explanation:

We can select the cells with values 7 and 8 that are colored above.

Constraints:

1 <= grid.length, grid[i].length <= 10
1 <= grid[i][j] <= 100

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

SOLUTION:

assert Solution().maxScore(grid = [[1,2,3],[3,2,1]]) == 5

assert Solution().maxScore(grid = [[1,2],[2,1]]) == 3

assert Solution().maxScore(grid = [[1]]) == 1

Answer: assert Solution().maxScore(grid = [[5,5,5],[5,5,5],[5,5,5]]) == 5

Answer: assert Solution().maxScore(grid = [[8,7,6],[8,3,2]]) == 15

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5],[5,4,3,2,1],[2,3,4,5,1]]) == 12

Answer: assert Solution().maxScore(grid = [[5,9,1],[9,5,1],[1,5,9]]) == 15

Answer: assert Solution().maxScore(grid = [[5,9,3],[4,5,6],[7,8,9]]) == 23

Answer: assert Solution().maxScore(grid = [[1,1,1],[1,1,1],[1,1,1]]) == 1

Answer: assert Solution().maxScore(grid = [[10,10,10],[10,10,10],[10,10,10]]) == 10

Answer: assert Solution().maxScore(grid = [[1,100,1],[100,1,100],[1,100,1]]) == 101

Answer: assert Solution().maxScore(grid = [[1,2,3],[3,2,1]]) == 5

Answer: assert Solution().maxScore(grid = [[1,2],[3,4],[5,6]]) == 12

Answer: assert Solution().maxScore(grid = [[9,8,7],[6,5,4],[3,2,1]]) == 18

Answer: assert Solution().maxScore(grid = [[1,2,3],[4,3,2],[1,1,1]]) == 8

Answer: assert Solution().maxScore(grid = [[5,6,7],[8,9,10],[11,12,13]]) == 30

Answer: assert Solution().maxScore(grid = [[1,2],[2,1]]) == 3

Answer: assert Solution().maxScore(grid = [[10,10],[10,10]]) == 10

Answer: assert Solution().maxScore(grid = [[10,20,30],[20,10,40],[30,40,50]]) == 120

Answer: assert Solution().maxScore(grid = [[5,9,1],[9,5,7],[6,7,4]]) == 22

Answer: assert Solution().maxScore(grid = [[1]]) == 1

Answer: assert Solution().maxScore(grid = [[10,20,30],[40,50,60],[70,80,90]]) == 180

Answer: assert Solution().maxScore(grid = [[1,2,3,4],[5,6,7,8],[9,10,11,12]]) == 24

Answer: assert Solution().maxScore(grid = [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]) == 63

Answer: assert Solution().maxScore(grid = [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]) == 45

Answer: assert Solution().maxScore(grid = [[5,5,5,5,5,5,5,5,5,5],[10,10,10,10,10,10,10,10,10,10],[15,15,15,15,15,15,15,15,15,15],[20,20,20,20,20,20,20,20,20,20]]) == 50

Answer: assert Solution().maxScore(grid = [[10, 20, 30], [30, 10, 20], [20, 30, 10], [10, 20, 30], [30, 20, 10]]) == 60

Answer: assert Solution().maxScore(grid = [[7,8,9,10,11],[6,7,8,9,10],[5,6,7,8,9],[4,5,6,7,8],[3,4,5,6,7]]) == 45

Answer: assert Solution().maxScore(grid = [[9, 8, 7, 6, 5], [5, 4, 3, 2, 1], [10, 11, 12, 13, 14], [14, 13, 12, 11, 10], [9, 8, 7, 6, 5]]) == 49

Answer: assert Solution().maxScore(grid = [[1,1,1,1],[2,2,2,2],[3,3,3,3],[4,4,4,4]]) == 10

Answer: assert Solution().maxScore(grid = [[99,100,98,97,96],[95,94,93,92,91],[90,89,88,87,86],[85,84,83,82,81],[80,79,78,77,76]]) == 450

Answer: assert Solution().maxScore(grid = [[100, 90, 80], [80, 90, 100], [70, 60, 50], [50, 60, 70], [40, 30, 20], [20, 30, 40]]) == 390

Answer: assert Solution().maxScore(grid = [[5,5,5,5,5,5],[6,6,6,6,6,6],[1,1,1,1,1,1],[7,7,7,7,7,7],[8,8,8,8,8,8],[9,9,9,9,9,9]]) == 36

Answer: assert Solution().maxScore(grid = [[9,8,7,6,5],[4,3,2,1,0],[9,1,8,2,7],[6,5,4,3,2],[1,9,8,7,6]]) == 34

Answer: assert Solution().maxScore(grid = [[1, 2, 3, 4], [4, 3, 2, 1], [5, 6, 7, 8], [8, 7, 6, 5], [9, 10, 11, 12], [12, 11, 10, 9]]) == 45

Answer: assert Solution().maxScore(grid = [[97,98,99,100],[93,94,95,96],[89,90,91,92],[85,86,87,88]]) == 376

Answer: assert Solution().maxScore(grid = [[5,5,5,5,5],[4,4,4,4,4],[3,3,3,3,3],[2,2,2,2,2],[1,1,1,1,1]]) == 15

Answer: assert Solution().maxScore(grid = [[5,2,7,1,9,3],[3,8,1,4,6,7],[9,1,8,3,4,2],[6,4,2,8,1,5],[7,9,3,6,2,8]]) == 35

Answer: assert Solution().maxScore(grid = [[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]]) == 15

Answer: assert Solution().maxScore(grid = [[1,5,9,12,4],[7,6,8,3,2],[10,11,13,1,5],[14,15,16,17,18],[19,20,21,22,23]]) == 74

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5,6,7,8,9,10],[10,9,8,7,6,5,4,3,2,1],[1,3,5,7,9,2,4,6,8,10],[10,8,6,4,2,9,7,5,3,1]]) == 34

Answer: assert Solution().maxScore(grid = [[10,9,8,7,6],[9,8,7,6,5],[8,7,6,5,4],[7,6,5,4,3],[6,5,4,3,2],[5,4,3,2,1]]) == 45

Answer: assert Solution().maxScore(grid = [[10,20,30],[30,20,10],[40,50,60],[60,50,40],[70,80,90]]) == 250

Answer: assert Solution().maxScore(grid = [[10,10,10,10],[20,20,20,20],[30,30,30,30],[40,40,40,40]]) == 100

Answer: assert Solution().maxScore(grid = [[10,20,30],[40,50,60],[70,80,90],[100,110,120],[130,140,150]]) == 450

Answer: assert Solution().maxScore(grid = [[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]]) == 15

Answer: assert Solution().maxScore(grid = [[5,10,15,20,25],[30,25,20,15,10],[5,30,15,20,25],[10,15,20,30,5],[15,20,25,10,30]]) == 100

Answer: assert Solution().maxScore(grid = [[10,20,30,40,50],[40,50,60,70,80],[50,60,70,80,90],[60,70,80,90,100],[70,80,90,100,110]]) == 430

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5,6,7,8,9,10],[10,9,8,7,6,5,4,3,2,1],[1,3,5,7,9,2,4,6,8,10]]) == 27

Answer: assert Solution().maxScore(grid = [[1, 2, 3], [4, 5, 6], [7, 8, 9], [10, 11, 12], [13, 14, 15]]) == 45

Answer: assert Solution().maxScore(grid = [[9, 2, 3, 8, 1], [4, 6, 7, 5, 3], [5, 8, 9, 1, 4], [2, 3, 4, 5, 6], [1, 7, 8, 9, 2]]) == 35

Answer: assert Solution().maxScore(grid = [[9,8,7,6,5],[4,3,2,1,10],[3,4,5,6,7],[2,1,8,9,10],[1,2,3,4,5]]) == 39

Answer: assert Solution().maxScore(grid = [[1,1,1,1,1,1],[1,2,2,2,2,1],[1,2,3,3,2,1],[1,2,3,4,3,1],[1,2,3,3,2,1],[1,1,1,1,1,1]]) == 10

Answer: assert Solution().maxScore(grid = [[100,99,98,97,96],[95,94,93,92,91],[90,89,88,87,86],[85,84,83,82,81],[80,79,78,77,76]]) == 450

Answer: assert Solution().maxScore(grid = [[10,20,30,40,50],[9,8,7,6,5],[1,2,3,4,5],[50,40,30,20,10]]) == 104

Answer: assert Solution().maxScore(grid = [[5,10,15,20,25,30],[30,25,20,15,10,5],[6,12,18,24,30,36],[36,30,24,18,12,6],[4,8,12,16,20,24],[24,20,16,12,8,4]]) == 153

Answer: assert Solution().maxScore(grid = [[10,20,30,40],[5,15,25,35],[1,11,21,31],[4,14,24,34]]) == 140

Answer: assert Solution().maxScore(grid = [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]) == 30

Answer: assert Solution().maxScore(grid = [[9,8,7,6,5,4,3,2,1],[1,2,3,4,5,6,7,8,9],[2,4,6,8,10,12,14,16,18],[3,6,9,12,15,18,21,24,27]]) == 62

Answer: assert Solution().maxScore(grid = [[10,20,30,40,50],[45,35,25,15,5],[1,2,3,4,5],[50,45,40,35,30]]) == 140

Answer: assert Solution().maxScore(grid = [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]) == 42

Answer: assert Solution().maxScore(grid = [[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],[5,5,5,5,5,5,5,5,5,5],[6,6,6,6,6,6,6,6,6,6],[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],[10,10,10,10,10,10,10,10,10,10]]) == 55

Answer: assert Solution().maxScore(grid = [[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

Answer: assert Solution().maxScore(grid = [[10,20,30,40,50],[50,40,30,20,10],[20,30,40,50,10],[10,20,30,40,50]]) == 140

Answer: assert Solution().maxScore(grid = [[50,50,50,50,50],[40,40,40,40,40],[30,30,30,30,30],[20,20,20,20,20],[10,10,10,10,10]]) == 150

Answer: assert Solution().maxScore(grid = [[1,1,1,2,2,3,3,3,4,4],[1,1,1,2,2,3,3,3,4,4],[1,1,1,2,2,3,3,3,4,4],[1,1,1,2,2,3,3,3,4,4]]) == 10

Answer: assert Solution().maxScore(grid = [[9,2,3,6,5],[4,8,7,3,9],[1,6,5,8,2],[7,4,1,9,5]]) == 30

Answer: assert Solution().maxScore(grid = [[5,3,7,9],[2,8,6,4],[10,1,11,12],[13,14,15,16]]) == 45

Answer: assert Solution().maxScore(grid = [[1,3,5,7,9],[9,7,5,3,1],[2,4,6,8,10],[10,8,6,4,2]]) == 34

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5],[2,3,4,5,6],[3,4,5,6,7],[4,5,6,7,8],[5,6,7,8,9],[6,7,8,9,10]]) == 45

Answer: assert Solution().maxScore(grid = [[90, 80, 70], [60, 50, 40], [30, 20, 10], [100, 95, 85], [80, 75, 70]]) == 360

Answer: assert Solution().maxScore(grid = [[10,9,8,7,6],[5,4,3,2,1],[9,8,7,6,5],[4,3,2,1,10]]) == 32

Answer: assert Solution().maxScore(grid = [[10,20,30,40,50],[50,40,30,20,10],[11,22,33,44,55],[55,44,33,22,11],[12,23,34,45,56]]) == 245

Answer: assert Solution().maxScore(grid = [[50,40,30,20,10],[11,21,31,41,51],[12,22,32,42,52],[13,23,33,43,53],[14,24,34,44,54]]) == 260

Answer: assert Solution().maxScore(grid = [[100,90,80,70,60,50],[50,60,70,80,90,100],[99,88,77,66,55,44],[11,22,33,44,55,66],[65,75,85,95,105,115]]) == 470

Answer: assert Solution().maxScore(grid = [[1,3,5,7,9],[2,4,6,8,10],[10,8,6,4,2],[9,7,5,3,1]]) == 34

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5],[2,3,4,5,6],[3,4,5,6,7],[4,5,6,7,8],[5,6,7,8,9]]) == 35

Answer: assert Solution().maxScore(grid = [[1,2,3,4],[2,3,4,5],[3,4,5,6],[4,5,6,7],[5,6,7,8]]) == 30

Answer: assert Solution().maxScore(grid = [[10,20,30,40,50],[50,40,30,20,10],[10,30,20,50,40],[20,10,50,40,30],[40,30,10,20,50]]) == 150

Answer: assert Solution().maxScore(grid = [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10], [11, 12], [13, 14], [15, 16], [17, 18], [19, 20]]) == 110

Answer: assert Solution().maxScore(grid = [[5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5], [5, 5, 5]]) == 5

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5],[5,4,3,2,1],[2,3,4,5,6],[6,5,4,3,2],[3,4,5,6,7]]) == 25

Answer: assert Solution().maxScore(grid = [[5,5,5,5,5,5],[4,4,4,4,4,4],[3,3,3,3,3,3],[2,2,2,2,2,2],[1,1,1,1,1,1]]) == 15

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

Answer: assert Solution().maxScore(grid = [[10, 20, 30, 40], [40, 30, 20, 10], [50, 60, 70, 80], [80, 70, 60, 50]]) == 220

Answer: assert Solution().maxScore(grid = [[10, 20, 30], [30, 20, 10], [40, 50, 60], [60, 50, 40], [70, 80, 90], [90, 80, 70]]) == 330

Answer: assert Solution().maxScore(grid = [[9,8,7,6,5,4,3,2,1],[1,2,3,4,5,6,7,8,9],[9,7,5,3,1,2,4,6,8],[2,4,6,8,10,9,7,5,3],[3,6,9,12,15,18,21,24,27]]) == 61

Answer: assert Solution().maxScore(grid = [[10,20,30,40,50],[50,40,30,20,10],[60,70,80,90,100],[100,90,80,70,60]]) == 280

Answer: assert Solution().maxScore(grid = [[9,8,7,6,5,4],[3,2,1,10,11,12],[13,14,15,16,17,18],[19,20,21,22,23,24],[25,26,27,28,29,30],[31,32,33,34,35,36]]) == 129

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5,6,7,8,9,10],[2,3,4,5,6,7,8,9,10,11],[3,4,5,6,7,8,9,10,11,12],[4,5,6,7,8,9,10,11,12,13]]) == 46

Answer: assert Solution().maxScore(grid = [[1,2],[2,1],[3,4],[4,3],[5,6],[6,5]]) == 21

Answer: assert Solution().maxScore(grid = [[1, 1, 2, 2, 3], [2, 2, 3, 3, 4], [3, 3, 4, 4, 5], [4, 4, 5, 5, 6], [5, 5, 6, 6, 7]]) == 25

Answer: assert Solution().maxScore(grid = [[1,1,2,2,3],[2,2,3,3,4],[3,3,4,4,5],[4,4,5,5,6],[5,5,6,6,7]]) == 25

Answer: assert Solution().maxScore(grid = [[1,2,3,4],[4,3,2,1],[2,3,4,1],[1,4,2,3]]) == 10

Answer: assert Solution().maxScore(grid = [[5, 15, 25, 35], [35, 25, 15, 5], [40, 50, 60, 70], [70, 60, 50, 40], [85, 95, 105, 115], [115, 105, 95, 85]]) == 410

Answer: assert Solution().maxScore(grid = [[3,3,3,3,3],[2,2,2,2,2],[1,1,1,1,1],[4,4,4,4,4],[5,5,5,5,5]]) == 15

Answer: assert Solution().maxScore(grid = [[1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [1, 3, 5, 7, 9, 2, 4, 6, 8, 10]]) == 27

Answer: assert Solution().maxScore(grid = [[1,2,3,4,5,6,7,8,9,10],[10,9,8,7,6,5,4,3,2,1],[2,3,4,5,6,7,8,9,10,1],[3,4,5,6,7,8,9,10,1,2]]) == 34

Answer: assert Solution().maxScore(grid = [[1, 2, 3, 4, 5], [5, 4, 3, 2, 1], [1, 1, 1, 1, 1], [2, 2, 2, 2, 2], [3, 3, 3, 3, 3]]) == 15

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 maxScore(self, grid: List[List[int]]) -> int:
        g = defaultdict(set)
        mx = 0
        for i, row in enumerate(grid):
            for x in row:
                g[x].add(i)
                mx = max(mx, x)
        m = len(grid)
        f = [[0] * (1 << m) for _ in range(mx + 1)]
        for i in range(1, mx + 1):
            for j in range(1 << m):
                f[i][j] = f[i - 1][j]
                for k in g[i]:
                    if j >> k & 1:
                        f[i][j] = max(f[i][j], f[i - 1][j ^ 1 << k] + i)
        return f[-1][-1]


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:
There are several pistons in an old car engine, and we want to calculate the maximum possible area under the pistons.
You are given:

An integer height, representing the maximum height a piston can reach.
An integer array positions, where positions[i] is the current position of piston i, which is equal to the current area under it.
A string directions, where directions[i] is the current moving direction of piston i, 'U' for up, and 'D' for down.

Each second:

Every piston moves in its current direction 1 unit. e.g., if the direction is up, positions[i] is incremented by 1.
If a piston has reached one of the ends, i.e., positions[i] == 0 or positions[i] == height, its direction will change.

Return the maximum possible area under all the pistons.

Example 1:

Input: height = 5, positions = [2,5], directions = "UD"
Output: 7
Explanation:
The current position of the pistons has the maximum possible area under it.

Example 2:

Input: height = 6, positions = [0,0,6,3], directions = "UUDU"
Output: 15
Explanation:
After 3 seconds, the pistons will be in positions [3, 3, 3, 6], which has the maximum possible area under it.

Constraints:

1 <= height <= 106
1 <= positions.length == directions.length <= 105
0 <= positions[i] <= height
directions[i] is either 'U' or 'D'.

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

SOLUTION:

Question:
PROBLEM:
There are several pistons in an old car engine, and we want to calculate the maximum possible area under the pistons.
You are given:

An integer height, representing the maximum height a piston can reach.
An integer array positions, where positions[i] is the current position of piston i, which is equal to the current area under it.
A string directions, where directions[i] is the current moving direction of piston i, 'U' for up, and 'D' for down.

Each second:

Every piston moves in its current direction 1 unit. e.g., if the direction is up, positions[i] is incremented by 1.
If a piston has reached one of the ends, i.e., positions[i] == 0 or positions[i] == height, its direction will change.

Return the maximum possible area under all the pistons.

Example 1:

Input: height = 5, positions = [2,5], directions = "UD"
Output: 7
Explanation:
The current position of the pistons has the maximum possible area under it.

Example 2:

Input: height = 6, positions = [0,0,6,3], directions = "UUDU"
Output: 15
Explanation:
After 3 seconds, the pistons will be in positions [3, 3, 3, 6], which has the maximum possible area under it.

Constraints:

1 <= height <= 106
1 <= positions.length == directions.length <= 105
0 <= positions[i] <= height
directions[i] is either 'U' or 'D'.

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

SOLUTION:

assert Solution().maxArea(height = 10, positions = [0,1,5,10], directions = "UUDD") == 24

assert Solution().maxArea(height = 100000, positions = [0,0,0,0,0], directions = "UUUUU") == 500000

assert Solution().maxArea(height = 20, positions = [10,10,10,10,10,10,10,10,10,10], directions = "UUUUUUUUUUUUUUUUUUUU") == 200

assert Solution().maxArea(height = 100, positions = [0,100,50,25,75], directions = "UDUDU") == 300

assert Solution().maxArea(height = 10, positions = [5,5,5,5], directions = 'UUDD') == 20

assert Solution().maxArea(height = 5, positions = [2,5], directions = "UD") == 7

assert Solution().maxArea(height = 1000, positions = [500,500,500,500], directions = "UUUU") == 4000

assert Solution().maxArea(height = 1, positions = [0,1], directions = "UD") == 1

assert Solution().maxArea(height = 100, positions = [50,50,50,50,50], directions = "UUUUU") == 500

assert Solution().maxArea(height = 6, positions = [0,0,6,3], directions = 'UUDU') == 15

assert Solution().maxArea(height = 3, positions = [0,3], directions = 'UD') == 3

assert Solution().maxArea(height = 7, positions = [7,6,5,4,3,2,1], directions = "DDDDDDD") == 37

assert Solution().maxArea(height = 1000000, positions = [0,1000000], directions = "UD") == 1000000

assert Solution().maxArea(height = 7, positions = [1,2,3,4,5,6,7], directions = "UUUUUUU") == 37

assert Solution().maxArea(height = 1, positions = [0,1], directions = 'DU') == 1

assert Solution().maxArea(height = 3, positions = [3,3,3], directions = "DDD") == 9

assert Solution().maxArea(height = 6, positions = [0,0,6,3], directions = "UUDU") == 15

assert Solution().maxArea(height = 10, positions = [10,10,10,10,10], directions = "DDDDD") == 50

assert Solution().maxArea(height = 10, positions = [1,2,3,4,5], directions = "UUUUU") == 44

assert Solution().maxArea(height = 5, positions = [2,5], directions = 'UD') == 7

assert Solution().maxArea(height = 10, positions = [0,5,10], directions = "UDD") == 20

assert Solution().maxArea(height = 200, positions = [100, 0, 200, 0, 100, 200, 0, 100, 200, 0, 100, 200, 0, 100, 200, 0, 100, 200, 0, 100], directions = 'UUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUU') == 2100

assert Solution().maxArea(height = 100, positions = [0, 0, 0, 0, 100, 100, 100, 100, 100], directions = 'UUUUUUUUU') == 500

assert Solution().maxArea(height = 50, positions = [25, 25, 25, 25, 25, 25, 25, 25, 25, 25], directions = 'UUUUUUUUUU') == 500

assert Solution().maxArea(height = 300, positions = [1, 299, 2, 298, 3, 297, 4, 296, 5, 295], directions = 'UUUUUUUUUU') == 1530

assert Solution().maxArea(height = 7, positions = [1, 1, 1, 6, 6, 6], directions = "UUUUUU") == 27

assert Solution().maxArea(height = 50, positions = [25, 25, 25, 25, 25, 25], directions = 'UUUUUU') == 300

assert Solution().maxArea(height = 9, positions = [0, 9, 3, 6, 3, 6, 3, 6, 3, 6], directions = 'UUDDUUDDUUDD') == 45

assert Solution().maxArea(height = 11, positions = [0, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], directions = 'UUDDDDDDDDDDD') == 121

assert Solution().maxArea(height = 400, positions = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 400, 400, 400, 400, 400], directions = 'UUUUUUUUUUDDDDDDDDDD') == 4000

assert Solution().maxArea(height = 100, positions = [0, 100, 50, 50, 0], directions = 'UUDDU') == 350

assert Solution().maxArea(height = 60, positions = [10, 50, 20, 40, 30, 30, 40, 20, 50, 10], directions = 'UUDDUUDDUU') == 320

assert Solution().maxArea(height = 20, positions = [0, 20, 0, 20, 0, 20, 0, 20, 0, 20], directions = 'UDUDUDUDUD') == 100

assert Solution().maxArea(height = 100, positions = [0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100], directions = 'UUDUDDUDUU') == 550

assert Solution().maxArea(height = 120, positions = [0, 30, 60, 90, 120, 0, 30, 60, 90, 120], directions = 'UUUUUUUUUU') == 840

assert Solution().maxArea(height = 75, positions = [25, 50, 75, 50, 25, 0, 75, 0, 50, 25], directions = 'UUDDUUDDUU') == 475

assert Solution().maxArea(height = 5000, positions = [1, 5000, 2500, 4999, 2, 4998], directions = 'UUDDUUDD') == 17500

assert Solution().maxArea(height = 1000, positions = [0, 1000, 500, 250, 750], directions = 'UUDDU') == 3000

assert Solution().maxArea(height = 100, positions = [90, 80, 70, 60, 50, 40, 30, 20, 10], directions = 'DDDDDDDDD') == 700

assert Solution().maxArea(height = 500, positions = [0, 250, 499, 1, 125, 375], directions = 'UDUDUU') == 1750

assert Solution().maxArea(height = 10, positions = [0, 10, 5, 5, 0, 10, 5, 5, 0, 10, 5], directions = 'UUDDUUDDUUDD') == 80

assert Solution().maxArea(height = 10, positions = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], directions = 'UUUUUUUUUUU') == 80

assert Solution().maxArea(height = 750, positions = [375, 375, 375, 375, 375, 375], directions = 'UUDDUU') == 3000

assert Solution().maxArea(height = 12, positions = [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], directions = 'DDDDDDDDDDDD') == 108

assert Solution().maxArea(height = 999, positions = [0, 999, 1, 998, 2, 997, 3, 996, 4, 995], directions = 'UDUDUDUDUD') == 4995

assert Solution().maxArea(height = 500, positions = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], directions = 'UUUUUUUUUU') == 5000

assert Solution().maxArea(height = 20, positions = [0, 20, 10, 10, 10, 10, 10], directions = 'UUDDDDD') == 120

assert Solution().maxArea(height = 10000, positions = [1, 2, 3, 9998, 9999, 10000], directions = 'UUUUUUDD') == 30009

assert Solution().maxArea(height = 100, positions = [99, 1, 98, 2, 97, 3, 96, 4], directions = "DDUUDDUU") == 404

assert Solution().maxArea(height = 7, positions = [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], directions = 'UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU') == 301

assert Solution().maxArea(height = 500, positions = [1, 499, 250, 100, 400], directions = "DDUUU") == 1698

assert Solution().maxArea(height = 9, positions = [0, 9, 1, 8, 2, 7, 3, 6, 4, 5], directions = 'UUUUUUUUUU') == 65

assert Solution().maxArea(height = 7, positions = [0, 1, 2, 3, 4, 5, 6, 7], directions = 'UUUUUUUU') == 40

assert Solution().maxArea(height = 100, positions = [10, 90, 50, 60, 40, 30], directions = 'UDDUDDUU') == 380

assert Solution().maxArea(height = 200, positions = [0, 0, 0, 0, 0, 200, 200, 200, 200, 200], directions = 'UUUUUDDDDD') == 1000

assert Solution().maxArea(height = 500, positions = [0, 250, 500, 250, 0, 250, 500, 250], directions = 'UDDUDDUDDU') == 2500

assert Solution().maxArea(height = 80, positions = [0, 16, 32, 48, 64, 80, 64, 48, 32, 16], directions = 'UUUUUUUUUU') == 592

assert Solution().maxArea(height = 500, positions = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], directions = 'UUUUUUUUUU') == 4750

assert Solution().maxArea(height = 50, positions = [10, 20, 30, 40, 50, 40, 30, 20, 10], directions = 'UUUUUUUUU') == 350

assert Solution().maxArea(height = 500, positions = [250, 250, 250, 250, 250, 250], directions = 'UUUUUU') == 3000

assert Solution().maxArea(height = 12, positions = [0, 12, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6], directions = 'UUDDDDDDDDDDD') == 144

assert Solution().maxArea(height = 20, positions = [0, 20, 10, 5, 15, 20, 0, 10, 5, 15], directions = 'UUDDUDDUUDDUUDDU') == 100

assert Solution().maxArea(height = 150, positions = [30, 60, 90, 120, 150, 120, 90, 60, 30], directions = 'UUUUUUUUU') == 1050

assert Solution().maxArea(height = 750, positions = [10, 740, 380, 670, 100, 650, 150, 550], directions = 'UDDUDDUDDU') == 3550

assert Solution().maxArea(height = 100, positions = [10, 90, 50, 60, 20], directions = "UUUUU") == 380

assert Solution().maxArea(height = 1000, positions = [0, 1000, 0, 1000, 0, 1000], directions = 'UUDDUU') == 3000

assert Solution().maxArea(height = 500, positions = [0, 500, 250, 125, 375], directions = 'UDDUU') == 1375

assert Solution().maxArea(height = 10000, positions = [5000, 5000, 5000, 5000, 5000], directions = 'UUUUU') == 50000

assert Solution().maxArea(height = 200, positions = [0, 200, 100, 100, 100, 100, 100, 100], directions = 'UUDDUUDD') == 1000

assert Solution().maxArea(height = 150, positions = [0, 150, 75, 25, 125, 50, 100, 100, 50, 125, 25, 75], directions = 'UUDDUDDUUDDUUDDUDDUUDDUUDDUUDD') == 950

assert Solution().maxArea(height = 250, positions = [50, 150, 250, 200, 100, 50], directions = 'UUUUUU') == 1100

assert Solution().maxArea(height = 150, positions = [50, 51, 52, 53, 54, 99, 98, 97, 96, 95], directions = 'UUUUUUUUUU') == 1275

assert Solution().maxArea(height = 100, positions = [10, 90, 50, 75], directions = 'UUDD') == 255

assert Solution().maxArea(height = 1500, positions = [0, 1500, 750, 375, 1125, 250, 1250], directions = 'UDDUDDUDDU') == 6000

assert Solution().maxArea(height = 100, positions = [1, 99, 2, 98, 3, 97], directions = 'UUUUUU') == 312

assert Solution().maxArea(height = 1000, positions = [0, 0, 0, 0, 0, 0], directions = 'UUDDUD') == 6000

assert Solution().maxArea(height = 100, positions = [0, 100, 50, 25, 75], directions = 'UUUUU') == 350

assert Solution().maxArea(height = 1200, positions = [0, 1200, 600, 300, 900, 150, 1050, 800], directions = 'UUDDUUDDUU') == 5800

assert Solution().maxArea(height = 1000, positions = [0, 1000, 500, 250, 750], directions = 'UUDDDD') == 3500

assert Solution().maxArea(height = 200, positions = [200, 0, 100, 100, 100, 100], directions = 'DDDUDU') == 600

assert Solution().maxArea(height = 600, positions = [300, 200, 100, 0, 600, 500, 400], directions = 'DDDDUUU') == 2400

assert Solution().maxArea(height = 15, positions = [0, 15, 0, 15, 0, 15, 0, 15, 0, 15, 0, 15, 0, 15, 0], directions = 'UDUDUDUDUDUDUDUDU') == 120

assert Solution().maxArea(height = 10, positions = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], directions = 'UUUUUUUUUU') == 75

assert Solution().maxArea(height = 800, positions = [400, 400, 400, 400, 400, 400, 400, 400], directions = 'UUUUUUUU') == 6400

assert Solution().maxArea(height = 2000, positions = [500, 1000, 1500, 2000, 0, 1000, 1500], directions = 'UUDDUUUU') == 9500

assert Solution().maxArea(height = 8, positions = [0, 8, 4, 4, 4, 4, 4, 4, 4, 4], directions = 'UUDDDDDDDD') == 72

assert Solution().maxArea(height = 8, positions = [1, 7, 2, 6, 3, 5, 4, 4], directions = 'UUUUUUUU') == 52

assert Solution().maxArea(height = 100, positions = [0, 100, 50, 25, 75], directions = 'UUDDU') == 300

assert Solution().maxArea(height = 100, positions = [10, 20, 30, 40, 50, 60, 70, 80, 90], directions = 'UUUUUUUUU') == 700

assert Solution().maxArea(height = 80, positions = [10, 70, 20, 60, 30, 50, 40], directions = 'UUUUUU') == 360

assert Solution().maxArea(height = 500, positions = [250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0], directions = 'UUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUU') == 5250

Answer: assert Solution().maxArea(height = 10, positions = [0,1,5,10], directions = "UUDD") == 24

Answer: assert Solution().maxArea(height = 100000, positions = [0,0,0,0,0], directions = "UUUUU") == 500000

Answer: assert Solution().maxArea(height = 20, positions = [10,10,10,10,10,10,10,10,10,10], directions = "UUUUUUUUUUUUUUUUUUUU") == 200

Answer: assert Solution().maxArea(height = 100, positions = [0,100,50,25,75], directions = "UDUDU") == 300

Answer: assert Solution().maxArea(height = 10, positions = [5,5,5,5], directions = 'UUDD') == 20

Answer: assert Solution().maxArea(height = 5, positions = [2,5], directions = "UD") == 7

Answer: assert Solution().maxArea(height = 1000, positions = [500,500,500,500], directions = "UUUU") == 4000

Answer: assert Solution().maxArea(height = 1, positions = [0,1], directions = "UD") == 1

Answer: assert Solution().maxArea(height = 100, positions = [50,50,50,50,50], directions = "UUUUU") == 500

Answer: assert Solution().maxArea(height = 6, positions = [0,0,6,3], directions = 'UUDU') == 15

Answer: assert Solution().maxArea(height = 3, positions = [0,3], directions = 'UD') == 3

Answer: assert Solution().maxArea(height = 7, positions = [7,6,5,4,3,2,1], directions = "DDDDDDD") == 37

Answer: assert Solution().maxArea(height = 1000000, positions = [0,1000000], directions = "UD") == 1000000

Answer: assert Solution().maxArea(height = 7, positions = [1,2,3,4,5,6,7], directions = "UUUUUUU") == 37

Answer: assert Solution().maxArea(height = 1, positions = [0,1], directions = 'DU') == 1

Answer: assert Solution().maxArea(height = 3, positions = [3,3,3], directions = "DDD") == 9

Answer: assert Solution().maxArea(height = 6, positions = [0,0,6,3], directions = "UUDU") == 15

Answer: assert Solution().maxArea(height = 10, positions = [10,10,10,10,10], directions = "DDDDD") == 50

Answer: assert Solution().maxArea(height = 10, positions = [1,2,3,4,5], directions = "UUUUU") == 44

Answer: assert Solution().maxArea(height = 5, positions = [2,5], directions = 'UD') == 7

Answer: assert Solution().maxArea(height = 10, positions = [0,5,10], directions = "UDD") == 20

Answer: assert Solution().maxArea(height = 200, positions = [100, 0, 200, 0, 100, 200, 0, 100, 200, 0, 100, 200, 0, 100, 200, 0, 100, 200, 0, 100], directions = 'UUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUU') == 2100

Answer: assert Solution().maxArea(height = 100, positions = [0, 0, 0, 0, 100, 100, 100, 100, 100], directions = 'UUUUUUUUU') == 500

Answer: assert Solution().maxArea(height = 50, positions = [25, 25, 25, 25, 25, 25, 25, 25, 25, 25], directions = 'UUUUUUUUUU') == 500

Answer: assert Solution().maxArea(height = 500000, positions = [250000, 250000, 250000, 250000, 250000], directions = 'UUUUU') == 2500000

Answer: assert Solution().maxArea(height = 550, positions = [150, 400, 250, 350, 100, 500, 50, 450, 300, 200], directions = 'UUUUUUUUUU') == 4250

Answer: assert Solution().maxArea(height = 300, positions = [1, 299, 2, 298, 3, 297, 4, 296, 5, 295], directions = 'UUUUUUUUUU') == 1530

Answer: assert Solution().maxArea(height = 7, positions = [1, 1, 1, 6, 6, 6], directions = "UUUUUU") == 27

Answer: assert Solution().maxArea(height = 50, positions = [25, 25, 25, 25, 25, 25], directions = 'UUUUUU') == 300

Answer: assert Solution().maxArea(height = 9, positions = [0, 9, 3, 6, 3, 6, 3, 6, 3, 6], directions = 'UUDDUUDDUUDD') == 45

Answer: assert Solution().maxArea(height = 11, positions = [0, 11, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], directions = 'UUDDDDDDDDDDD') == 121

Answer: assert Solution().maxArea(height = 400, positions = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 400, 400, 400, 400, 400], directions = 'UUUUUUUUUUDDDDDDDDDD') == 4000

Answer: assert Solution().maxArea(height = 100, positions = [0, 100, 50, 50, 0], directions = 'UUDDU') == 350

Answer: assert Solution().maxArea(height = 60, positions = [10, 50, 20, 40, 30, 30, 40, 20, 50, 10], directions = 'UUDDUUDDUU') == 320

Answer: assert Solution().maxArea(height = 150, positions = [75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75, 75], directions = 'UUUUUUUUUUUUUUUUUU') == 2400

Answer: assert Solution().maxArea(height = 20, positions = [0, 20, 0, 20, 0, 20, 0, 20, 0, 20], directions = 'UDUDUDUDUD') == 100

Answer: assert Solution().maxArea(height = 100, positions = [0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100], directions = 'UUDUDDUDUU') == 550

Answer: assert Solution().maxArea(height = 120, positions = [0, 30, 60, 90, 120, 0, 30, 60, 90, 120], directions = 'UUUUUUUUUU') == 840

Answer: assert Solution().maxArea(height = 25, positions = [25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], directions = 'DDDDDDDDDDDDDDDDDDDDDDDD') == 456

Answer: assert Solution().maxArea(height = 200, positions = [100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100, 100], directions = 'UUUUUUUUUUUUUUUUUUUU') == 4000

Answer: assert Solution().maxArea(height = 75, positions = [25, 50, 75, 50, 25, 0, 75, 0, 50, 25], directions = 'UUDDUUDDUU') == 475

Answer: assert Solution().maxArea(height = 5000, positions = [1, 5000, 2500, 4999, 2, 4998], directions = 'UUDDUUDD') == 17500

Answer: assert Solution().maxArea(height = 1000, positions = [0, 1000, 500, 250, 750], directions = 'UUDDU') == 3000

Answer: assert Solution().maxArea(height = 100, positions = [90, 80, 70, 60, 50, 40, 30, 20, 10], directions = 'DDDDDDDDD') == 700

Answer: assert Solution().maxArea(height = 1200, positions = [100, 1100, 200, 1000, 300, 900, 400, 800, 500, 700], directions = 'UUUUUUUUUU') == 9000

Answer: assert Solution().maxArea(height = 500, positions = [0, 250, 499, 1, 125, 375], directions = 'UDUDUU') == 1750

Answer: assert Solution().maxArea(height = 10, positions = [0, 10, 5, 5, 0, 10, 5, 5, 0, 10, 5], directions = 'UUDDUUDDUUDD') == 80

Answer: assert Solution().maxArea(height = 10, positions = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], directions = 'UUUUUUUUUUU') == 80

Answer: assert Solution().maxArea(height = 750, positions = [375, 375, 375, 375, 375, 375], directions = 'UUDDUU') == 3000

Answer: assert Solution().maxArea(height = 12, positions = [11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0], directions = 'DDDDDDDDDDDD') == 108

Answer: assert Solution().maxArea(height = 999, positions = [0, 999, 1, 998, 2, 997, 3, 996, 4, 995], directions = 'UDUDUDUDUD') == 4995

Answer: assert Solution().maxArea(height = 500, positions = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], directions = 'UUUUUUUUUU') == 5000

Answer: assert Solution().maxArea(height = 20, positions = [0, 20, 10, 10, 10, 10, 10], directions = 'UUDDDDD') == 120

Answer: assert Solution().maxArea(height = 2500, positions = [0, 1250, 2500, 625, 1875, 312, 2187, 437, 1562, 1937], directions = 'UUDDUUDDUUDD') == 14813

Answer: assert Solution().maxArea(height = 10000, positions = [1, 2, 3, 9998, 9999, 10000], directions = 'UUUUUUDD') == 30009

Answer: assert Solution().maxArea(height = 100, positions = [99, 1, 98, 2, 97, 3, 96, 4], directions = "DDUUDDUU") == 404

Answer: assert Solution().maxArea(height = 7, positions = [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], directions = 'UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU') == 301

Answer: assert Solution().maxArea(height = 500, positions = [1, 499, 250, 100, 400], directions = "DDUUU") == 1698

Answer: assert Solution().maxArea(height = 9, positions = [0, 9, 1, 8, 2, 7, 3, 6, 4, 5], directions = 'UUUUUUUUUU') == 65

Answer: assert Solution().maxArea(height = 7, positions = [0, 1, 2, 3, 4, 5, 6, 7], directions = 'UUUUUUUU') == 40

Answer: assert Solution().maxArea(height = 100, positions = [10, 90, 50, 60, 40, 30], directions = 'UDDUDDUU') == 380

Answer: assert Solution().maxArea(height = 100, positions = [1, 99, 2, 98, 3, 97, 4, 96, 5, 95, 6, 94, 7, 93, 8, 92, 9, 91, 10, 90], directions = 'UUUUUUUUUUUUUUUUUUUU') == 1110

Answer: assert Solution().maxArea(height = 200, positions = [0, 0, 0, 0, 0, 200, 200, 200, 200, 200], directions = 'UUUUUDDDDD') == 1000

Answer: assert Solution().maxArea(height = 120, positions = [0, 30, 60, 90, 120, 90, 60, 30, 0, 30, 60, 90, 120, 90, 60, 30, 0, 30, 60, 90], directions = 'UUUUUUUUUUUUUUUUUUUU') == 1800

Answer: assert Solution().maxArea(height = 100, positions = [1, 1, 1, 1, 1, 99, 99, 99, 99, 99, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50], directions = 'UUUUUUUUUUUUUUUUUUUU') == 1510

Answer: assert Solution().maxArea(height = 500, positions = [0, 250, 500, 250, 0, 250, 500, 250], directions = 'UDDUDDUDDU') == 2500

Answer: assert Solution().maxArea(height = 1000000, positions = [0, 1000000, 500000, 500000], directions = "UUDD") == 3000000

Answer: assert Solution().maxArea(height = 80, positions = [0, 16, 32, 48, 64, 80, 64, 48, 32, 16], directions = 'UUUUUUUUUU') == 592

Answer: assert Solution().maxArea(height = 500, positions = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], directions = 'UUUUUUUUUU') == 4750

Answer: assert Solution().maxArea(height = 450, positions = [100, 350, 200, 300, 150, 250, 50, 400, 450, 0], directions = 'UUUUUUUUUU') == 3250

Answer: assert Solution().maxArea(height = 300, positions = [150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150, 150], directions = 'UUUUUUUUUUUU') == 3600

Answer: assert Solution().maxArea(height = 50, positions = [10, 20, 30, 40, 50, 40, 30, 20, 10], directions = 'UUUUUUUUU') == 350

Answer: assert Solution().maxArea(height = 1000, positions = [500, 500, 500, 500, 500, 500, 500, 500, 500, 500], directions = 'UUUUUUUUUU') == 10000

Answer: assert Solution().maxArea(height = 500, positions = [250, 250, 250, 250, 250, 250], directions = 'UUUUUU') == 3000

Answer: assert Solution().maxArea(height = 12, positions = [0, 12, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6], directions = 'UUDDDDDDDDDDD') == 144

Answer: assert Solution().maxArea(height = 20, positions = [0, 20, 10, 5, 15, 20, 0, 10, 5, 15], directions = 'UUDDUDDUUDDUUDDU') == 100

Answer: assert Solution().maxArea(height = 150, positions = [30, 60, 90, 120, 150, 120, 90, 60, 30], directions = 'UUUUUUUUU') == 1050

Answer: assert Solution().maxArea(height = 750, positions = [10, 740, 380, 670, 100, 650, 150, 550], directions = 'UDDUDDUDDU') == 3550

Answer: assert Solution().maxArea(height = 100, positions = [10, 90, 50, 60, 20], directions = "UUUUU") == 380

Answer: assert Solution().maxArea(height = 10000, positions = [1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10000], directions = 'DDDDDDDDDD') == 75000

Answer: assert Solution().maxArea(height = 1000, positions = [0, 1000, 0, 1000, 0, 1000], directions = 'UUDDUU') == 3000

Answer: assert Solution().maxArea(height = 500, positions = [0, 500, 250, 125, 375], directions = 'UDDUU') == 1375

Answer: assert Solution().maxArea(height = 10000, positions = [5000, 5000, 5000, 5000, 5000], directions = 'UUUUU') == 50000

Answer: assert Solution().maxArea(height = 200, positions = [0, 200, 100, 100, 100, 100, 100, 100], directions = 'UUDDUUDD') == 1000

Answer: assert Solution().maxArea(height = 150, positions = [0, 150, 75, 25, 125, 50, 100, 100, 50, 125, 25, 75], directions = 'UUDDUDDUUDDUUDDUDDUUDDUUDDUUDD') == 950

Answer: assert Solution().maxArea(height = 250, positions = [50, 150, 250, 200, 100, 50], directions = 'UUUUUU') == 1100

Answer: assert Solution().maxArea(height = 150, positions = [50, 51, 52, 53, 54, 99, 98, 97, 96, 95], directions = 'UUUUUUUUUU') == 1275

Answer: assert Solution().maxArea(height = 100, positions = [10, 90, 50, 75], directions = 'UUDD') == 255

Answer: assert Solution().maxArea(height = 1500, positions = [0, 1500, 750, 375, 1125, 250, 1250], directions = 'UDDUDDUDDU') == 6000

Answer: assert Solution().maxArea(height = 300, positions = [0, 300, 150, 150, 150, 150, 150, 150, 150, 150], directions = 'UUDDUUDDUU') == 1500

Answer: assert Solution().maxArea(height = 100, positions = [1, 99, 2, 98, 3, 97], directions = 'UUUUUU') == 312

Answer: assert Solution().maxArea(height = 1000, positions = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], directions = 'UUUUUUUUUU') == 7500

Answer: assert Solution().maxArea(height = 1000, positions = [0, 0, 0, 0, 0, 0], directions = 'UUDDUD') == 6000

Answer: assert Solution().maxArea(height = 100, positions = [0, 100, 50, 25, 75], directions = 'UUUUU') == 350

Answer: assert Solution().maxArea(height = 1200, positions = [0, 1200, 600, 300, 900, 150, 1050, 800], directions = 'UUDDUUDDUU') == 5800

Answer: assert Solution().maxArea(height = 1000, positions = [0, 1000, 500, 250, 750], directions = 'UUDDDD') == 3500

Answer: assert Solution().maxArea(height = 75, positions = [10, 20, 30, 40, 50, 60, 70, 65, 55, 45, 35, 25, 15, 5], directions = 'UUUUUUUUUUUUUU') == 805

Answer: assert Solution().maxArea(height = 200, positions = [200, 0, 100, 100, 100, 100], directions = 'DDDUDU') == 600

Answer: assert Solution().maxArea(height = 600, positions = [300, 200, 100, 0, 600, 500, 400], directions = 'DDDDUUU') == 2400

Answer: assert Solution().maxArea(height = 15, positions = [0, 15, 0, 15, 0, 15, 0, 15, 0, 15, 0, 15, 0, 15, 0], directions = 'UDUDUDUDUDUDUDUDU') == 120

Answer: assert Solution().maxArea(height = 10, positions = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], directions = 'UUUUUUUUUU') == 75

Answer: assert Solution().maxArea(height = 800, positions = [400, 400, 400, 400, 400, 400, 400, 400], directions = 'UUUUUUUU') == 6400

Answer: assert Solution().maxArea(height = 2000, positions = [500, 1000, 1500, 2000, 0, 1000, 1500], directions = 'UUDDUUUU') == 9500

Answer: assert Solution().maxArea(height = 20, positions = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10], directions = 'UUUUUUUUUUUUUUUUUUUU') == 400

Answer: assert Solution().maxArea(height = 8, positions = [0, 8, 4, 4, 4, 4, 4, 4, 4, 4], directions = 'UUDDDDDDDD') == 72

Answer: assert Solution().maxArea(height = 8, positions = [1, 7, 2, 6, 3, 5, 4, 4], directions = 'UUUUUUUU') == 52

Answer: assert Solution().maxArea(height = 200, positions = [100, 100, 100, 100, 100, 100, 100, 100, 100, 100], directions = 'UUUUUUUUUU') == 2000

Answer: assert Solution().maxArea(height = 100, positions = [0, 100, 50, 25, 75], directions = 'UUDDU') == 300

Answer: assert Solution().maxArea(height = 100, positions = [10, 20, 30, 40, 50, 60, 70, 80, 90], directions = 'UUUUUUUUU') == 700

Answer: assert Solution().maxArea(height = 80, positions = [10, 70, 20, 60, 30, 50, 40], directions = 'UUUUUU') == 360

Answer: assert Solution().maxArea(height = 50, positions = [1, 49, 2, 48, 3, 47, 4, 46, 5, 45, 6, 44, 7, 43, 8, 42, 9, 41, 10, 40, 11, 39, 12, 38, 13, 37, 14, 36, 15, 35, 16, 34, 17, 33, 18, 32, 19, 31, 20, 30, 21, 29, 22, 28, 23, 27, 24, 26], directions = 'UUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUU') == 1800

Answer: assert Solution().maxArea(height = 500, positions = [0, 100, 200, 300, 400, 500, 400, 300, 200, 100], directions = 'UUDDUUDDUU') == 2700

Answer: assert Solution().maxArea(height = 500, positions = [250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0, 500, 250, 0], directions = 'UUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUUDDUUDDUUDDUUUUDDUUDDUUUU') == 5250

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 maxArea(self, height: int, positions: List[int], directions: str) -> int:
        delta = defaultdict(int)
        diff = res = 0
        for pos, dir in zip(positions, directions):
            res += pos
            if dir == "U":
                diff += 1
                delta[height - pos] -= 2
                delta[height * 2 - pos] += 2
            else:
                diff -= 1
                delta[pos] += 2
                delta[height + pos] -= 2
        ans = res
        pre = 0
        for cur, d in sorted(delta.items()):
            res += (cur - pre) * diff
            pre = cur
            diff += d
            ans = max(ans, res)
        return ans

class Solution:
    def maxArea(self, height: int, positions: List[int], directions: str) -> int:


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

PROBLEM:
You are given an array of integers start and an integer d, representing n intervals [start[i], start[i] + d].
You are asked to choose n integers where the ith integer must belong to the ith interval. The score of the chosen integers is defined as the minimum absolute difference between any two integers that have been chosen.
Return the maximum possible score of the chosen integers.

Example 1:

Input: start = [6,0,3], d = 2
Output: 4
Explanation:
The maximum possible score can be obtained by choosing integers: 8, 0, and 4. The score of these chosen integers is min(|8 - 0|, |8 - 4|, |0 - 4|) which equals 4.

Example 2:

Input: start = [2,6,13,13], d = 5
Output: 5
Explanation:
The maximum possible score can be obtained by choosing integers: 2, 7, 13, and 18. The score of these chosen integers is min(|2 - 7|, |2 - 13|, |2 - 18|, |7 - 13|, |7 - 18|, |13 - 18|) which equals 5.

Constraints:

2 <= start.length <= 105
0 <= start[i] <= 109
0 <= d <= 109

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

SOLUTION:

Question:
PROBLEM:
You are given an array of integers start and an integer d, representing n intervals [start[i], start[i] + d].
You are asked to choose n integers where the ith integer must belong to the ith interval. The score of the chosen integers is defined as the minimum absolute difference between any two integers that have been chosen.
Return the maximum possible score of the chosen integers.

Example 1:

Input: start = [6,0,3], d = 2
Output: 4
Explanation:
The maximum possible score can be obtained by choosing integers: 8, 0, and 4. The score of these chosen integers is min(|8 - 0|, |8 - 4|, |0 - 4|) which equals 4.

Example 2:

Input: start = [2,6,13,13], d = 5
Output: 5
Explanation:
The maximum possible score can be obtained by choosing integers: 2, 7, 13, and 18. The score of these chosen integers is min(|2 - 7|, |2 - 13|, |2 - 18|, |7 - 13|, |7 - 18|, |13 - 18|) which equals 5.

Constraints:

2 <= start.length <= 105
0 <= start[i] <= 109
0 <= d <= 109

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

SOLUTION:

assert Solution().maxPossibleScore(start = [5,10,15,20,25], d = 4) == 6

assert Solution().maxPossibleScore(start = [1,10,20,30], d = 9) == 12

assert Solution().maxPossibleScore(start = [10,20,30,40,50], d = 15) == 13

assert Solution().maxPossibleScore(start = [100,200,300,400,500], d = 50) == 112

assert Solution().maxPossibleScore(start = [2,6,13,13], d = 5) == 5

assert Solution().maxPossibleScore(start = [1,1,1,1], d = 10) == 3

assert Solution().maxPossibleScore(start = [5,5,5,5,5], d = 0) == 0

assert Solution().maxPossibleScore(start = [10,20,30,40,50,60], d = 15) == 13

assert Solution().maxPossibleScore(start = [1,2,3,4,5], d = 1) == 1

assert Solution().maxPossibleScore(start = [1,5,9,14], d = 3) == 5

assert Solution().maxPossibleScore(start = [0,1,2,3,4,5,6,7,8,9], d = 1) == 1

assert Solution().maxPossibleScore(start = [6,0,3], d = 2) == 4

assert Solution().maxPossibleScore(start = [1,3,5,7,9], d = 2) == 2

assert Solution().maxPossibleScore(start = [100,200,300,400], d = 50) == 116

assert Solution().maxPossibleScore(start = [0,0,0,0], d = 10) == 3

assert Solution().maxPossibleScore(start = [1,1,1], d = 10) == 5

assert Solution().maxPossibleScore(start = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], d = 0) == 1

assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 1) == 0

assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0], d = 50) == 10

assert Solution().maxPossibleScore(start = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], d = 100) == 11

assert Solution().maxPossibleScore(start = [1, 5, 10, 20, 25, 30], d = 10) == 7

assert Solution().maxPossibleScore(start = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], d = 9) == 2

assert Solution().maxPossibleScore(start = [1, 5, 9, 13, 17, 21], d = 3) == 4

assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 100) == 11

assert Solution().maxPossibleScore(start = [5, 15, 25, 35, 45, 55, 65], d = 20) == 13

assert Solution().maxPossibleScore(start = [1, 5, 9, 13, 17, 21], d = 4) == 4

assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 0) == 0

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

Answer: assert Solution().maxPossibleScore(start = [5,10,15,20,25], d = 4) == 6

Answer: assert Solution().maxPossibleScore(start = [1,10,20,30], d = 9) == 12

Answer: assert Solution().maxPossibleScore(start = [10,20,30,40,50], d = 15) == 13

Answer: assert Solution().maxPossibleScore(start = [100,200,300,400,500], d = 50) == 112

Answer: assert Solution().maxPossibleScore(start = [2,6,13,13], d = 5) == 5

Answer: assert Solution().maxPossibleScore(start = [1,1,1,1], d = 10) == 3

Answer: assert Solution().maxPossibleScore(start = [5,5,5,5,5], d = 0) == 0

Answer: assert Solution().maxPossibleScore(start = [10,20,30,40,50,60], d = 15) == 13

Answer: assert Solution().maxPossibleScore(start = [1,2,3,4,5], d = 1) == 1

Answer: assert Solution().maxPossibleScore(start = [1,5,9,14], d = 3) == 5

Answer: assert Solution().maxPossibleScore(start = [0,1,2,3,4,5,6,7,8,9], d = 1) == 1

Answer: assert Solution().maxPossibleScore(start = [0,1000000000], d = 999999999) == 1999999999

Answer: assert Solution().maxPossibleScore(start = [6,0,3], d = 2) == 4

Answer: assert Solution().maxPossibleScore(start = [1,3,5,7,9], d = 2) == 2

Answer: assert Solution().maxPossibleScore(start = [100,200,300,400], d = 50) == 116

Answer: assert Solution().maxPossibleScore(start = [0,0,0,0], d = 10) == 3

Answer: assert Solution().maxPossibleScore(start = [1,1,1], d = 10) == 5

Answer: assert Solution().maxPossibleScore(start = [0,1000000000], d = 1000000000) == 2000000000

Answer: assert Solution().maxPossibleScore(start = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], d = 0) == 1

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

Answer: assert Solution().maxPossibleScore(start = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], d = 50) == 105

Answer: assert Solution().maxPossibleScore(start = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], d = 100) == 111

Answer: assert Solution().maxPossibleScore(start = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024], d = 1) == 2

Answer: assert Solution().maxPossibleScore(start = [1000000000, 1000000000, 1000000000, 1000000000, 1000000000], d = 1) == 0

Answer: assert Solution().maxPossibleScore(start = [1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190], d = 10) == 10

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 1) == 0

Answer: assert Solution().maxPossibleScore(start = [10, 25, 40, 55, 70, 85, 100, 115, 130, 145, 160, 175, 190, 205, 220, 235, 250, 265, 280, 295], d = 15) == 15

Answer: assert Solution().maxPossibleScore(start = [0, 1000000000, 500000000, 750000000, 250000000], d = 1000000000) == 500000000

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 100) == 7

Answer: assert Solution().maxPossibleScore(start = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384], d = 1) == 2

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

Answer: assert Solution().maxPossibleScore(start = [0, 1000000000, 2000000000, 3000000000, 4000000000, 5000000000], d = 1000000000) == 1200000000

Answer: assert Solution().maxPossibleScore(start = [100, 200, 300, 400, 500, 600, 700], d = 50) == 108

Answer: assert Solution().maxPossibleScore(start = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500], d = 10) == 100

Answer: assert Solution().maxPossibleScore(start = [1, 3, 6, 10, 15, 21, 28, 36, 45, 55], d = 1) == 3

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

Answer: assert Solution().maxPossibleScore(start = [1, 1000000000, 2, 1000000001, 3, 1000000002], d = 100000000) == 50000001

Answer: assert Solution().maxPossibleScore(start = [0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30], d = 1) == 2

Answer: assert Solution().maxPossibleScore(start = [100,200,300,400,500,600,700,800,900,1000,1100,1200,1300,1400,1500,1600,1700,1800,1900,2000], d = 100) == 105

Answer: assert Solution().maxPossibleScore(start = [1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000], d = 1) == 10

Answer: assert Solution().maxPossibleScore(start = [0, 10, 20, 30, 40, 50, 60, 70, 80, 90], d = 10) == 11

Answer: assert Solution().maxPossibleScore(start = [0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105], d = 1) == 2

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 1000000000) == 52631578

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0], d = 50) == 10

Answer: assert Solution().maxPossibleScore(start = [1000000000, 2000000000, 3000000000], d = 1000000000) == 1500000000

Answer: assert Solution().maxPossibleScore(start = [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], d = 1) == 1

Answer: assert Solution().maxPossibleScore(start = [1000000000, 2000000000, 3000000000, 4000000000, 5000000000, 6000000000], d = 1000000000) == 1200000000

Answer: assert Solution().maxPossibleScore(start = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], d = 100) == 11

Answer: assert Solution().maxPossibleScore(start = [500000000, 1000000000, 1500000000, 2000000000, 2500000000, 3000000000, 3500000000, 4000000000, 4500000000, 5000000000], d = 500000000) == 555555555

Answer: assert Solution().maxPossibleScore(start = [1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000], d = 1000000000) == 137499999

Answer: assert Solution().maxPossibleScore(start = [1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325], d = 1) == 3

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 1000000000) == 111111111

Answer: assert Solution().maxPossibleScore(start = [1, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100], d = 9) == 10

Answer: assert Solution().maxPossibleScore(start = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], d = 1000000000) == 111111111

Answer: assert Solution().maxPossibleScore(start = [1000000000, 999999900, 999999800, 999999700, 999999600, 999999500], d = 50) == 110

Answer: assert Solution().maxPossibleScore(start = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], d = 10) == 11

Answer: assert Solution().maxPossibleScore(start = [1000000000, 1000000000, 1000000000, 1000000000, 1000000000], d = 0) == 0

Answer: assert Solution().maxPossibleScore(start = [1000000000, 2000000000, 3000000000, 4000000000, 5000000000], d = 1000000000) == 1250000000

Answer: assert Solution().maxPossibleScore(start = [1, 10, 100, 1000, 10000], d = 999) == 549

Answer: assert Solution().maxPossibleScore(start = [0, 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], d = 2) == 1

Answer: assert Solution().maxPossibleScore(start = [1, 5, 10, 20, 25, 30], d = 10) == 7

Answer: assert Solution().maxPossibleScore(start = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200], d = 5) == 10

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 1) == 0

Answer: assert Solution().maxPossibleScore(start = [5, 11, 17, 23, 29, 35, 41, 47, 53, 59, 65, 71, 77, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 197, 203, 209, 215, 221, 227, 233, 239, 245, 251, 257, 263, 269, 275, 281, 287, 293, 299], d = 6) == 6

Answer: assert Solution().maxPossibleScore(start = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9], d = 9) == 2

Answer: assert Solution().maxPossibleScore(start = [5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105, 115, 125], d = 10) == 10

Answer: assert Solution().maxPossibleScore(start = [2, 6, 10, 14, 18, 22, 26, 30, 34, 38], d = 3) == 4

Answer: assert Solution().maxPossibleScore(start = [1, 5, 9, 13, 17, 21], d = 3) == 4

Answer: assert Solution().maxPossibleScore(start = [0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100], d = 15) == 11

Answer: assert Solution().maxPossibleScore(start = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], d = 2) == 2

Answer: assert Solution().maxPossibleScore(start = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512], d = 10) == 5

Answer: assert Solution().maxPossibleScore(start = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40], d = 2) == 2

Answer: assert Solution().maxPossibleScore(start = [5, 15, 25, 35, 45, 55, 65, 75, 85, 95], d = 10) == 11

Answer: assert Solution().maxPossibleScore(start = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99], d = 2) == 2

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 100) == 11

Answer: assert Solution().maxPossibleScore(start = [1000000, 2000000, 3000000, 4000000, 5000000, 6000000, 7000000, 8000000, 9000000, 10000000], d = 100000) == 1011111

Answer: assert Solution().maxPossibleScore(start = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150], d = 10) == 10

Answer: assert Solution().maxPossibleScore(start = [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49], d = 4) == 4

Answer: assert Solution().maxPossibleScore(start = [5, 15, 25, 35, 45, 55, 65], d = 20) == 13

Answer: assert Solution().maxPossibleScore(start = [1, 5, 10, 20, 30, 40, 50, 60, 70, 80], d = 10) == 9

Answer: assert Solution().maxPossibleScore(start = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], d = 1) == 2

Answer: assert Solution().maxPossibleScore(start = [1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45, 49, 53, 57], d = 3) == 4

Answer: assert Solution().maxPossibleScore(start = [100,200,300,400,500,600,700,800,900,1000,1100,1200,1300,1400,1500], d = 50) == 103

Answer: assert Solution().maxPossibleScore(start = [1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000, 1000000000], d = 0) == 0

Answer: assert Solution().maxPossibleScore(start = [0,1000000000,2000000000,3000000000], d = 999999999) == 1333333333

Answer: assert Solution().maxPossibleScore(start = [1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37,39], d = 1) == 2

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

Answer: assert Solution().maxPossibleScore(start = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0], d = 1000000000) == 52631578

Answer: assert Solution().maxPossibleScore(start = [1, 5, 9, 13, 17, 21], d = 4) == 4

Answer: assert Solution().maxPossibleScore(start = [10, 12, 15, 18, 22, 26, 30, 35, 40, 45], d = 5) == 4

Answer: assert Solution().maxPossibleScore(start = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], d = 0) == 0

Answer: assert Solution().maxPossibleScore(start = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], d = 100) == 7

Answer: assert Solution().maxPossibleScore(start = [0, 1000000000, 500000000, 750000000], d = 1000000000) == 666666666

Answer: assert Solution().maxPossibleScore(start = [1,5,9,13,17,21,25,29,33,37,41,45,49,53,57,61,65,69,73,77], d = 2) == 4

Answer: assert Solution().maxPossibleScore(start = [100, 250, 400, 550, 700, 850, 1000, 1150, 1300, 1450, 1600, 1750], d = 50) == 154

Answer: assert Solution().maxPossibleScore(start = [1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1000000000,1000000000], d = 0) == 0

Answer: assert Solution().maxPossibleScore(start = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50], d = 10) == 6

Answer: assert Solution().maxPossibleScore(start = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150], d = 5) == 10

Answer: assert Solution().maxPossibleScore(start = [10,20,30,40,50,60,70,80,90,100], d = 10) == 11

Answer: assert Solution().maxPossibleScore(start = [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], d = 10) == 0

Answer: assert Solution().maxPossibleScore(start = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29], d = 1) == 2

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

Answer: assert Solution().maxPossibleScore(start = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], d = 2) == 2

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

Answer: assert Solution().maxPossibleScore(start = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], d = 5) == 10

Answer: assert Solution().maxPossibleScore(start = [5,10,15,20,25,30,35,40,45,50], d = 0) == 5

Answer: assert Solution().maxPossibleScore(start = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39], d = 1) == 2

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 maxPossibleScore(self, start: List[int], d: int) -> int:
        def check(mi: int) -> bool:
            last = -inf
            for st in start:
                if last + mi > st + d:
                    return False
                last = max(st, last + mi)
            return True

        start.sort()
        l, r = 0, start[-1] + d - start[0]
        while l < r:
            mid = (l + r + 1) >> 1
            if check(mid):
                l = mid
            else:
                r = mid - 1
        return l