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

PROBLEM:
You are given two integer arrays arr and brr of length n, and an integer k. You can perform the following operations on arr any number of times:

Split arr into any number of contiguous subarrays and rearrange these subarrays in any order. This operation has a fixed cost of k.

Choose any element in arr and add or subtract a positive integer x to it. The cost of this operation is x.

Return the minimum total cost to make arr equal to brr.

Example 1:

Input: arr = [-7,9,5], brr = [7,-2,-5], k = 2
Output: 13
Explanation:

Split arr into two contiguous subarrays: [-7] and [9, 5] and rearrange them as [9, 5, -7], with a cost of 2.
Subtract 2 from element arr[0]. The array becomes [7, 5, -7]. The cost of this operation is 2.
Subtract 7 from element arr[1]. The array becomes [7, -2, -7]. The cost of this operation is 7.
Add 2 to element arr[2]. The array becomes [7, -2, -5]. The cost of this operation is 2.

The total cost to make the arrays equal is 2 + 2 + 7 + 2 = 13.

Example 2:

Input: arr = [2,1], brr = [2,1], k = 0
Output: 0
Explanation:
Since the arrays are already equal, no operations are needed, and the total cost is 0.

Constraints:

1 <= arr.length == brr.length <= 105
0 <= k <= 2 * 1010
-105 <= arr[i] <= 105
-105 <= brr[i] <= 105

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

SOLUTION:

Question:
PROBLEM:
You are given two integer arrays arr and brr of length n, and an integer k. You can perform the following operations on arr any number of times:

Split arr into any number of contiguous subarrays and rearrange these subarrays in any order. This operation has a fixed cost of k.

Choose any element in arr and add or subtract a positive integer x to it. The cost of this operation is x.

Return the minimum total cost to make arr equal to brr.

Example 1:

Input: arr = [-7,9,5], brr = [7,-2,-5], k = 2
Output: 13
Explanation:

Split arr into two contiguous subarrays: [-7] and [9, 5] and rearrange them as [9, 5, -7], with a cost of 2.
Subtract 2 from element arr[0]. The array becomes [7, 5, -7]. The cost of this operation is 2.
Subtract 7 from element arr[1]. The array becomes [7, -2, -7]. The cost of this operation is 7.
Add 2 to element arr[2]. The array becomes [7, -2, -5]. The cost of this operation is 2.

The total cost to make the arrays equal is 2 + 2 + 7 + 2 = 13.

Example 2:

Input: arr = [2,1], brr = [2,1], k = 0
Output: 0
Explanation:
Since the arrays are already equal, no operations are needed, and the total cost is 0.

Constraints:

1 <= arr.length == brr.length <= 105
0 <= k <= 2 * 1010
-105 <= arr[i] <= 105
-105 <= brr[i] <= 105

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

SOLUTION:

assert Solution().minCost(arr = [2,1], brr = [2,1], k = 0) == 0

assert Solution().minCost(arr = [0,0,0], brr = [1,1,1], k = 1) == 3

assert Solution().minCost(arr = [1,2,3], brr = [3,2,1], k = 5) == 4

assert Solution().minCost(arr = [1,2,3], brr = [3,2,1], k = 1) == 1

Answer: assert Solution().minCost(arr = [0,0,0], brr = [0,0,0], k = 10) == 0

Answer: assert Solution().minCost(arr = [2,1], brr = [2,1], k = 0) == 0

Answer: assert Solution().minCost(arr = [0,0,0], brr = [0,0,0], k = 100) == 0

Answer: assert Solution().minCost(arr = [1,2,3], brr = [3,2,1], k = 10) == 4

Answer: assert Solution().minCost(arr = [0,0,0], brr = [1,1,1], k = 1) == 3

Answer: assert Solution().minCost(arr = [100000,-100000,50000], brr = [50000,-100000,100000], k = 5) == 5

Answer: assert Solution().minCost(arr = [10,20,30], brr = [30,20,10], k = 5) == 5

Answer: assert Solution().minCost(arr = [100000,-100000], brr = [-100000,100000], k = 200000) == 200000

Answer: assert Solution().minCost(arr = [-7,9,5], brr = [7,-2,-5], k = 2) == 13

Answer: assert Solution().minCost(arr = [1,2,3], brr = [3,2,1], k = 5) == 4

Answer: assert Solution().minCost(arr = [100000,-100000,50000], brr = [50000,100000,-100000], k = 1000) == 1000

Answer: assert Solution().minCost(arr = [1,1,1,1,1], brr = [1,1,1,1,1], k = 1000000000) == 0

Answer: assert Solution().minCost(arr = [-10,-20,-30], brr = [30,20,10], k = 100) == 120

Answer: assert Solution().minCost(arr = [1,-1,1,-1], brr = [-1,1,-1,1], k = 3) == 3

Answer: assert Solution().minCost(arr = [1,2,3], brr = [3,2,1], k = 1) == 1

Answer: assert Solution().minCost(arr = [1,3,5,7,9], brr = [9,7,5,3,1], k = 5) == 5

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 200000) == 0

Answer: assert Solution().minCost(arr = [10, 20, 30, 40, 50], brr = [5, 15, 25, 35, 45], k = 10) == 25

Answer: assert Solution().minCost(arr = [10, -10, 20, -20, 30, -30], brr = [-30, 30, -20, 20, -10, 10], k = 100) == 100

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 100000) == 0

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

Answer: assert Solution().minCost(arr = [1, 1, 2, 2, 3, 3], brr = [3, 3, 2, 2, 1, 1], k = 1) == 1

Answer: assert Solution().minCost(arr = [5, 15, 25, 35, 45], brr = [45, 35, 25, 15, 5], k = 20) == 20

Answer: assert Solution().minCost(arr = [10000, 10000, 10000], brr = [10000, 10000, 10000], k = 0) == 0

Answer: assert Solution().minCost(arr = [100000, -100000, 50000, -50000], brr = [-50000, 50000, -100000, 100000], k = 50000) == 50000

Answer: assert Solution().minCost(arr = [5, -3, 8, 10, -1], brr = [-1, 8, 5, -3, 10], k = 5) == 5

Answer: assert Solution().minCost(arr = [-10, 10, -10, 10, -10], brr = [10, -10, 10, -10, 10], k = 7) == 27

Answer: assert Solution().minCost(arr = [5, -5, 10, -10, 15, -15], brr = [-15, 15, -10, 10, -5, 5], k = 20) == 20

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 1000000000) == 0

Answer: assert Solution().minCost(arr = [10, 20, 30], brr = [30, 20, 10], k = 1000000000) == 40

Answer: assert Solution().minCost(arr = [5, 10, 15, 20, 25], brr = [25, 20, 15, 10, 5], k = 0) == 0

Answer: assert Solution().minCost(arr = [10, 20, 30, 40, 50], brr = [50, 40, 30, 20, 10], k = 5) == 5

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], k = 0) == 10

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], brr = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], k = 50) == 50

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500], brr = [500, 400, 300, 200, 100], k = 10000) == 1200

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1], brr = [2, 2, 2, 2, 2], k = 100000) == 5

Answer: assert Solution().minCost(arr = [5, 3, 8, 6, 2], brr = [2, 6, 3, 8, 5], k = 5) == 5

Answer: assert Solution().minCost(arr = [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5], brr = [5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5], k = 20) == 20

Answer: assert Solution().minCost(arr = [100000, -100000, 50000, -50000, 0], brr = [0, 50000, -100000, 100000, -50000], k = 1000) == 1000

Answer: assert Solution().minCost(arr = [5, -5, 5, -5, 5, -5, 5, -5, 5, -5], brr = [-5, 5, -5, 5, -5, 5, -5, 5, -5, 5], k = 2) == 2

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], brr = [10, 1, 2, 3, 4, 5, 6, 7, 8, 9], k = 50000) == 18

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

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5], brr = [5, 4, 3, 2, 1], k = 1) == 1

Answer: assert Solution().minCost(arr = [1, -2, 3, -4, 5, -6], brr = [-6, 5, -4, 3, -2, 1], k = 10) == 10

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

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

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], brr = [1000, 900, 800, 700, 600, 500, 400, 300, 200, 100], k = 100) == 100

Answer: assert Solution().minCost(arr = [-10, -20, -30, -40, -50], brr = [-50, -40, -30, -20, -10], k = 50) == 50

Answer: assert Solution().minCost(arr = [100000, -100000, 50000, -50000, 0], brr = [0, 50000, -50000, 100000, -100000], k = 1000) == 1000

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [2, 2, 2, 2, 2, 2, 2, 2, 2, 2], k = 1) == 10

Answer: assert Solution().minCost(arr = [-5, -10, -15, -20, -25], brr = [-25, -20, -15, -10, -5], k = 2) == 2

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], brr = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1], k = 20) == 20

Answer: assert Solution().minCost(arr = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], brr = [100, 90, 80, 70, 60, 50, 40, 30, 20, 10], k = 0) == 0

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

Answer: assert Solution().minCost(arr = [1000, -1000, 2000, -2000, 3000, -3000], brr = [3000, -3000, 2000, -2000, 1000, -1000], k = 100) == 100

Answer: assert Solution().minCost(arr = [100000, -100000, 50000, -50000], brr = [-50000, 50000, -100000, 100000], k = 5000) == 5000

Answer: assert Solution().minCost(arr = [50000, 40000, 30000, 20000, 10000], brr = [10000, 20000, 30000, 40000, 50000], k = 100000) == 100000

Answer: assert Solution().minCost(arr = [100000, 90000, 80000, 70000, 60000, 50000, 40000, 30000, 20000, 10000], brr = [10000, 20000, 30000, 40000, 50000, 60000, 70000, 80000, 90000, 100000], k = 5000) == 5000

Answer: assert Solution().minCost(arr = [10, 20, 30, 40, 50], brr = [50, 40, 30, 20, 10], k = 100) == 100

Answer: assert Solution().minCost(arr = [-100000, 100000, -100000, 100000, -100000, 100000], brr = [100000, -100000, 100000, -100000, 100000, -100000], k = 1) == 1

Answer: assert Solution().minCost(arr = [-50000, 50000, -50000, 50000, -50000], brr = [50000, -50000, 50000, -50000, 50000], k = 10000) == 110000

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500], brr = [500, 400, 300, 200, 100], k = 100) == 100

Answer: assert Solution().minCost(arr = [-10, -20, -30, -40, -50], brr = [-50, -40, -30, -20, -10], k = 10) == 10

Answer: assert Solution().minCost(arr = [1000, 2000, 3000, 4000, 5000], brr = [5000, 4000, 3000, 2000, 1000], k = 1000) == 1000

Answer: assert Solution().minCost(arr = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], brr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 1) == 55

Answer: assert Solution().minCost(arr = [5, 15, 25, 35, 45], brr = [45, 35, 25, 15, 5], k = 50) == 50

Answer: assert Solution().minCost(arr = [1, 3, 5, 7, 9], brr = [9, 7, 5, 3, 1], k = 5) == 5

Answer: assert Solution().minCost(arr = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], brr = [19, 17, 15, 13, 11, 9, 7, 5, 3, 1], k = 1000000000) == 100

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500], brr = [500, 100, 300, 200, 400], k = 20) == 20

Answer: assert Solution().minCost(arr = [-1, 0, 1, -2, 0, 2, -3, 0, 3, -4, 0, 4], brr = [4, 0, -4, 3, 0, -3, 2, 0, -2, 1, 0, -1], k = 2) == 2

Answer: assert Solution().minCost(arr = [100000, -100000, 50000, -50000], brr = [-50000, 50000, -100000, 100000], k = 10000) == 10000

Answer: assert Solution().minCost(arr = [100000, 100000, 100000, 100000, 100000], brr = [-100000, -100000, -100000, -100000, -100000], k = 10000) == 1000000

Answer: assert Solution().minCost(arr = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], brr = [100, 1, 2, 3, 4, 5, 6, 7, 8, 90], k = 50) == 374

Answer: assert Solution().minCost(arr = [-100, -200, -300, -400, -500, -600, -700, -800, -900, -1000], brr = [-1000, -900, -800, -700, -600, -500, -400, -300, -200, -100], k = 1) == 1

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

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

Answer: assert Solution().minCost(arr = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], brr = [19, 17, 15, 13, 11, 9, 7, 5, 3, 1], k = 2) == 2

Answer: assert Solution().minCost(arr = [-50000, -40000, -30000, -20000, -10000], brr = [-10000, -20000, -30000, -40000, -50000], k = 2000) == 2000

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

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], brr = [1000, 900, 800, 700, 600, 500, 400, 300, 200, 100], k = 50) == 50

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

Answer: assert Solution().minCost(arr = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], brr = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10], k = 0) == 50

Answer: assert Solution().minCost(arr = [100000, -100000, 100000, -100000, 100000], brr = [100000, -100000, 100000, -100000, 100000], k = 200000) == 0

Answer: assert Solution().minCost(arr = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], brr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 100) == 25

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

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

Answer: assert Solution().minCost(arr = [10000, 10000, 10000, 10000, 10000], brr = [-10000, -10000, -10000, -10000, -10000], k = 5000) == 100000

Answer: assert Solution().minCost(arr = [10, -20, 30, -40, 50], brr = [50, 30, 10, -20, -40], k = 5) == 5

Answer: assert Solution().minCost(arr = [-50000, -50000, -50000, -50000], brr = [50000, 50000, 50000, 50000], k = 50000) == 400000

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5], brr = [5, 4, 3, 2, 1], k = 1000) == 12

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], brr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 1000000000) == 5445

Answer: assert Solution().minCost(arr = [5, 5, 5, 5, 5], brr = [5, 5, 5, 5, 5], k = 1000000000) == 0

Answer: assert Solution().minCost(arr = [10, 20, 30, 40, 50], brr = [50, 40, 30, 20, 10], k = 0) == 0

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10], k = 5) == 90

Answer: assert Solution().minCost(arr = [-100, 0, 100], brr = [100, 0, -100], k = 100) == 100

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500], brr = [500, 400, 300, 200, 100], k = 1000) == 1000

Answer: assert Solution().minCost(arr = [50000, 40000, 30000, 20000, 10000], brr = [10000, 20000, 30000, 40000, 50000], k = 100) == 100

Answer: assert Solution().minCost(arr = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], brr = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 100000) == 0

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

Answer: assert Solution().minCost(arr = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], brr = [19, 17, 15, 13, 11, 9, 7, 5, 3, 1], k = 7) == 7

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], brr = [0, 0, 0, 0, 0, 0, 0, 0, 0, 0], k = 2) == 55

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5], brr = [5, 4, 3, 2, 1], k = 0) == 0

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

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

Answer: assert Solution().minCost(arr = [1, 3, 5, 7, 9], brr = [9, 7, 5, 3, 1], k = 1000000000) == 24

Answer: assert Solution().minCost(arr = [-10, -20, -30, -40, -50], brr = [-50, -40, -30, -20, -10], k = 20) == 20

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

Answer: assert Solution().minCost(arr = [100000, 100000, 100000, 100000], brr = [1, 1, 1, 1], k = 10000000000) == 399996

Answer: assert Solution().minCost(arr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], brr = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], k = 100) == 0

Answer: assert Solution().minCost(arr = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], brr = [19, 17, 15, 13, 11, 9, 7, 5, 3, 1], k = 1) == 1

Answer: assert Solution().minCost(arr = [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5], brr = [5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5], k = 1) == 1

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1], brr = [5, 5, 5, 5, 5], k = 3) == 20

Answer: assert Solution().minCost(arr = [-5, -5, -5, -5, -5], brr = [-5, -5, -5, -5, -5], k = 1) == 0

Answer: assert Solution().minCost(arr = [-5, 15, -25, 35, -45], brr = [-45, 35, -25, 15, -5], k = 0) == 0

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], k = 10) == 0

Answer: assert Solution().minCost(arr = [-10, 20, -30, 40], brr = [40, -30, 20, -10], k = 5) == 5

Answer: assert Solution().minCost(arr = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], brr = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10], k = 1) == 90

Answer: assert Solution().minCost(arr = [-50000, -40000, -30000, -20000, -10000], brr = [-10000, -20000, -30000, -40000, -50000], k = 10000) == 10000

Answer: assert Solution().minCost(arr = [1000000000, -1000000000, 0], brr = [0, 1000000000, -1000000000], k = 2000000000) == 2000000000

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

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

Answer: assert Solution().minCost(arr = [1, -1, 2, -2, 3, -3], brr = [3, -3, 2, -2, 1, -1], k = 10) == 8

Answer: assert Solution().minCost(arr = [100, 200, 300, 400, 500], brr = [500, 400, 300, 200, 100], k = 1000000000) == 1200

Answer: assert Solution().minCost(arr = [-5, -10, -15, -20, -25], brr = [-25, -20, -15, -10, -5], k = 10) == 10

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

Answer: assert Solution().minCost(arr = [-1, -2, -3, -4, -5, -6, -7, -8, -9, -10], brr = [-10, -9, -8, -7, -6, -5, -4, -3, -2, -1], k = 1) == 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 minCost(self, arr: List[int], brr: List[int], k: int) -> int:
        c1 = sum(abs(a - b) for a, b in zip(arr, brr))
        arr.sort()
        brr.sort()
        c2 = k + sum(abs(a - b) for a, b in zip(arr, brr))
        return min(c1, c2)


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 three integers m, n, and k.
There is a rectangular grid of size m × n containing k identical pieces. Return the sum of Manhattan distances between every pair of pieces over all valid arrangements of pieces.
A valid arrangement is a placement of all k pieces on the grid with at most one piece per cell.
Since the answer may be very large, return it modulo 109 + 7.
The Manhattan Distance between two cells (xi, yi) and (xj, yj) is |xi - xj| + |yi - yj|.

Example 1:

Input: m = 2, n = 2, k = 2
Output: 8
Explanation:
The valid arrangements of pieces on the board are:

In the first 4 arrangements, the Manhattan distance between the two pieces is 1.
In the last 2 arrangements, the Manhattan distance between the two pieces is 2.

Thus, the total Manhattan distance across all valid arrangements is 1 + 1 + 1 + 1 + 2 + 2 = 8.

Example 2:

Input: m = 1, n = 4, k = 3
Output: 20
Explanation:
The valid arrangements of pieces on the board are:

The first and last arrangements have a total Manhattan distance of 1 + 1 + 2 = 4.
The middle two arrangements have a total Manhattan distance of 1 + 2 + 3 = 6.

The total Manhattan distance between all pairs of pieces across all arrangements is 4 + 6 + 6 + 4 = 20.

Constraints:

1 <= m, n <= 105
2 <= m * n <= 105
2 <= k <= m * n

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

SOLUTION:

Question:
PROBLEM:
You are given three integers m, n, and k.
There is a rectangular grid of size m × n containing k identical pieces. Return the sum of Manhattan distances between every pair of pieces over all valid arrangements of pieces.
A valid arrangement is a placement of all k pieces on the grid with at most one piece per cell.
Since the answer may be very large, return it modulo 109 + 7.
The Manhattan Distance between two cells (xi, yi) and (xj, yj) is |xi - xj| + |yi - yj|.

Example 1:

Input: m = 2, n = 2, k = 2
Output: 8
Explanation:
The valid arrangements of pieces on the board are:

In the first 4 arrangements, the Manhattan distance between the two pieces is 1.
In the last 2 arrangements, the Manhattan distance between the two pieces is 2.

Thus, the total Manhattan distance across all valid arrangements is 1 + 1 + 1 + 1 + 2 + 2 = 8.

Example 2:

Input: m = 1, n = 4, k = 3
Output: 20
Explanation:
The valid arrangements of pieces on the board are:

The first and last arrangements have a total Manhattan distance of 1 + 1 + 2 = 4.
The middle two arrangements have a total Manhattan distance of 1 + 2 + 3 = 6.

The total Manhattan distance between all pairs of pieces across all arrangements is 4 + 6 + 6 + 4 = 20.

Constraints:

1 <= m, n <= 105
2 <= m * n <= 105
2 <= k <= m * n

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

SOLUTION:

assert Solution().distanceSum(m = 5, n = 5, k = 5) == 1771000

assert Solution().distanceSum(m = 10, n = 10, k = 20) == 80781589

assert Solution().distanceSum(m = 1, n = 4, k = 3) == 20

assert Solution().distanceSum(m = 10, n = 10, k = 10) == 800580356

assert Solution().distanceSum(m = 10, n = 10, k = 5) == 19167965

assert Solution().distanceSum(m = 3, n = 3, k = 4) == 1512

assert Solution().distanceSum(m = 2, n = 2, k = 2) == 8

assert Solution().distanceSum(m = 5, n = 5, k = 25) == 1000

assert Solution().distanceSum(m = 50, n = 50, k = 200) == 461891397

assert Solution().distanceSum(m = 20, n = 20, k = 100) == 642676804

assert Solution().distanceSum(m = 20, n = 20, k = 40) == 112638724

assert Solution().distanceSum(m = 4, n = 11, k = 22) == 286084962

assert Solution().distanceSum(m = 7, n = 7, k = 10) == 742720485

assert Solution().distanceSum(m = 7, n = 8, k = 20) == 830928354

assert Solution().distanceSum(m = 50, n = 50, k = 250) == 133218151

assert Solution().distanceSum(m = 25, n = 35, k = 500) == 268900107

assert Solution().distanceSum(m = 8, n = 5, k = 10) == 293801805

assert Solution().distanceSum(m = 8, n = 7, k = 30) == 245055662

assert Solution().distanceSum(m = 25, n = 20, k = 10) == 467663753

assert Solution().distanceSum(m = 3, n = 9, k = 8) == 248648400

assert Solution().distanceSum(m = 15, n = 15, k = 50) == 144305104

assert Solution().distanceSum(m = 2, n = 8, k = 5) == 145600

assert Solution().distanceSum(m = 100, n = 1, k = 50) == 596684460

assert Solution().distanceSum(m = 2, n = 10, k = 15) == 6511680

assert Solution().distanceSum(m = 15, n = 15, k = 20) == 814749998

assert Solution().distanceSum(m = 7, n = 7, k = 49) == 5488

assert Solution().distanceSum(m = 5, n = 20, k = 20) == 350976988

assert Solution().distanceSum(m = 3, n = 30, k = 15) == 694221074

assert Solution().distanceSum(m = 2, n = 10, k = 9) == 24186240

assert Solution().distanceSum(m = 50, n = 2, k = 49) == 464438831

assert Solution().distanceSum(m = 100, n = 1, k = 10) == 642930774

assert Solution().distanceSum(m = 15, n = 10, k = 50) == 608985873

assert Solution().distanceSum(m = 12, n = 15, k = 120) == 614641576

assert Solution().distanceSum(m = 2, n = 20, k = 20) == 162144738

assert Solution().distanceSum(m = 6, n = 9, k = 30) == 84183825

assert Solution().distanceSum(m = 9, n = 4, k = 15) == 395647069

assert Solution().distanceSum(m = 10, n = 10, k = 15) == 633547079

assert Solution().distanceSum(m = 20, n = 15, k = 100) == 132852497

assert Solution().distanceSum(m = 8, n = 9, k = 40) == 353742539

assert Solution().distanceSum(m = 3, n = 7, k = 6) == 2713200

assert Solution().distanceSum(m = 9, n = 9, k = 40) == 14076899

assert Solution().distanceSum(m = 10, n = 5, k = 15) == 518666205

assert Solution().distanceSum(m = 25, n = 25, k = 300) == 287498383

assert Solution().distanceSum(m = 50, n = 50, k = 125) == 203227794

assert Solution().distanceSum(m = 20, n = 5, k = 10) == 725438

assert Solution().distanceSum(m = 7, n = 7, k = 14) == 685863003

assert Solution().distanceSum(m = 50, n = 2, k = 100) == 85800

assert Solution().distanceSum(m = 4, n = 6, k = 12) == 594914320

assert Solution().distanceSum(m = 15, n = 7, k = 50) == 236557592

assert Solution().distanceSum(m = 1, n = 1000, k = 10) == 122738391

assert Solution().distanceSum(m = 1, n = 100, k = 50) == 596684460

assert Solution().distanceSum(m = 4, n = 12, k = 20) == 723849533

assert Solution().distanceSum(m = 9, n = 11, k = 99) == 32340

assert Solution().distanceSum(m = 50, n = 50, k = 100) == 31872683

assert Solution().distanceSum(m = 6, n = 7, k = 21) == 662311702

assert Solution().distanceSum(m = 12, n = 5, k = 35) == 152704514

assert Solution().distanceSum(m = 6, n = 6, k = 10) == 753633765

assert Solution().distanceSum(m = 7, n = 8, k = 15) == 927055315

assert Solution().distanceSum(m = 20, n = 15, k = 120) == 745855507

assert Solution().distanceSum(m = 8, n = 9, k = 45) == 569458134

assert Solution().distanceSum(m = 9, n = 12, k = 54) == 458241022

assert Solution().distanceSum(m = 3, n = 8, k = 24) == 1012

assert Solution().distanceSum(m = 20, n = 5, k = 50) == 543733780

assert Solution().distanceSum(m = 50, n = 2, k = 10) == 281508913

assert Solution().distanceSum(m = 7, n = 3, k = 10) == 52907400

assert Solution().distanceSum(m = 10, n = 10, k = 50) == 434987024

assert Solution().distanceSum(m = 6, n = 6, k = 18) == 982764729

assert Solution().distanceSum(m = 5, n = 10, k = 25) == 329316644

assert Solution().distanceSum(m = 6, n = 5, k = 20) == 931360310

assert Solution().distanceSum(m = 7, n = 5, k = 15) == 136117652

assert Solution().distanceSum(m = 15, n = 20, k = 100) == 132852497

assert Solution().distanceSum(m = 100, n = 100, k = 5) == 666666283

assert Solution().distanceSum(m = 1000, n = 1, k = 10) == 122738391

assert Solution().distanceSum(m = 15, n = 15, k = 100) == 268949671

assert Solution().distanceSum(m = 50, n = 50, k = 500) == 833366694

assert Solution().distanceSum(m = 75, n = 75, k = 300) == 770041881

assert Solution().distanceSum(m = 20, n = 5, k = 15) == 541933847

assert Solution().distanceSum(m = 20, n = 25, k = 100) == 883287723

assert Solution().distanceSum(m = 7, n = 6, k = 20) == 753814653

assert Solution().distanceSum(m = 30, n = 30, k = 90) == 482659936

Answer: assert Solution().distanceSum(m = 5, n = 5, k = 5) == 1771000

Answer: assert Solution().distanceSum(m = 100, n = 100, k = 10) == 384963906

Answer: assert Solution().distanceSum(m = 10, n = 10, k = 20) == 80781589

Answer: assert Solution().distanceSum(m = 1, n = 4, k = 3) == 20

Answer: assert Solution().distanceSum(m = 10, n = 10, k = 10) == 800580356

Answer: assert Solution().distanceSum(m = 10, n = 10, k = 5) == 19167965

Answer: assert Solution().distanceSum(m = 3, n = 3, k = 4) == 1512

Answer: assert Solution().distanceSum(m = 2, n = 2, k = 2) == 8

Answer: assert Solution().distanceSum(m = 5, n = 5, k = 25) == 1000

Answer: assert Solution().distanceSum(m = 50, n = 50, k = 200) == 461891397

Answer: assert Solution().distanceSum(m = 20, n = 20, k = 100) == 642676804

Answer: assert Solution().distanceSum(m = 1000, n = 1, k = 500) == 349233848

Answer: assert Solution().distanceSum(m = 20, n = 20, k = 40) == 112638724

Answer: assert Solution().distanceSum(m = 100, n = 100, k = 100) == 981249729

Answer: assert Solution().distanceSum(m = 4, n = 11, k = 22) == 286084962

Answer: assert Solution().distanceSum(m = 7, n = 7, k = 10) == 742720485

Answer: assert Solution().distanceSum(m = 7, n = 8, k = 20) == 830928354

Answer: assert Solution().distanceSum(m = 50, n = 50, k = 250) == 133218151

Answer: assert Solution().distanceSum(m = 100, n = 100, k = 200) == 917140960

Answer: assert Solution().distanceSum(m = 25, n = 35, k = 500) == 268900107

Answer: assert Solution().distanceSum(m = 8, n = 5, k = 10) == 293801805

Answer: assert Solution().distanceSum(m = 8, n = 7, k = 30) == 245055662

Answer: assert Solution().distanceSum(m = 25, n = 20, k = 10) == 467663753

Answer: assert Solution().distanceSum(m = 3, n = 9, k = 8) == 248648400

Answer: assert Solution().distanceSum(m = 15, n = 15, k = 50) == 144305104

Answer: assert Solution().distanceSum(m = 2, n = 8, k = 5) == 145600

Answer: assert Solution().distanceSum(m = 100, n = 1, k = 50) == 596684460

Answer: assert Solution().distanceSum(m = 2, n = 10, k = 15) == 6511680

Answer: assert Solution().distanceSum(m = 15, n = 15, k = 20) == 814749998

Answer: assert Solution().distanceSum(m = 7, n = 7, k = 49) == 5488

Answer: assert Solution().distanceSum(m = 5, n = 20, k = 20) == 350976988

Answer: assert Solution().distanceSum(m = 3, n = 30, k = 15) == 694221074

Answer: assert Solution().distanceSum(m = 2, n = 10, k = 9) == 24186240

Answer: assert Solution().distanceSum(m = 50, n = 2, k = 49) == 464438831

Answer: assert Solution().distanceSum(m = 100, n = 1, k = 10) == 642930774

Answer: assert Solution().distanceSum(m = 15, n = 10, k = 50) == 608985873

Answer: assert Solution().distanceSum(m = 12, n = 15, k = 120) == 614641576

Answer: assert Solution().distanceSum(m = 100, n = 100, k = 1000) == 934487950

Answer: assert Solution().distanceSum(m = 2, n = 20, k = 20) == 162144738

Answer: assert Solution().distanceSum(m = 6, n = 9, k = 30) == 84183825

Answer: assert Solution().distanceSum(m = 9, n = 4, k = 15) == 395647069

Answer: assert Solution().distanceSum(m = 10, n = 10, k = 15) == 633547079

Answer: assert Solution().distanceSum(m = 20, n = 15, k = 100) == 132852497

Answer: assert Solution().distanceSum(m = 8, n = 9, k = 40) == 353742539

Answer: assert Solution().distanceSum(m = 3, n = 7, k = 6) == 2713200

Answer: assert Solution().distanceSum(m = 9, n = 9, k = 40) == 14076899

Answer: assert Solution().distanceSum(m = 10, n = 5, k = 15) == 518666205

Answer: assert Solution().distanceSum(m = 25, n = 25, k = 300) == 287498383

Answer: assert Solution().distanceSum(m = 50, n = 50, k = 125) == 203227794

Answer: assert Solution().distanceSum(m = 20, n = 5, k = 10) == 725438

Answer: assert Solution().distanceSum(m = 7, n = 7, k = 14) == 685863003

Answer: assert Solution().distanceSum(m = 50, n = 2, k = 100) == 85800

Answer: assert Solution().distanceSum(m = 4, n = 6, k = 12) == 594914320

Answer: assert Solution().distanceSum(m = 15, n = 7, k = 50) == 236557592

Answer: assert Solution().distanceSum(m = 1, n = 1000, k = 10) == 122738391

Answer: assert Solution().distanceSum(m = 1, n = 100, k = 50) == 596684460

Answer: assert Solution().distanceSum(m = 4, n = 12, k = 20) == 723849533

Answer: assert Solution().distanceSum(m = 9, n = 11, k = 99) == 32340

Answer: assert Solution().distanceSum(m = 50, n = 50, k = 100) == 31872683

Answer: assert Solution().distanceSum(m = 6, n = 7, k = 21) == 662311702

Answer: assert Solution().distanceSum(m = 12, n = 5, k = 35) == 152704514

Answer: assert Solution().distanceSum(m = 6, n = 6, k = 10) == 753633765

Answer: assert Solution().distanceSum(m = 7, n = 8, k = 15) == 927055315

Answer: assert Solution().distanceSum(m = 20, n = 15, k = 120) == 745855507

Answer: assert Solution().distanceSum(m = 8, n = 9, k = 45) == 569458134

Answer: assert Solution().distanceSum(m = 100, n = 100, k = 50) == 106535598

Answer: assert Solution().distanceSum(m = 9, n = 12, k = 54) == 458241022

Answer: assert Solution().distanceSum(m = 100, n = 100, k = 10000) == 332999979

Answer: assert Solution().distanceSum(m = 3, n = 8, k = 24) == 1012

Answer: assert Solution().distanceSum(m = 20, n = 5, k = 50) == 543733780

Answer: assert Solution().distanceSum(m = 50, n = 2, k = 10) == 281508913

Answer: assert Solution().distanceSum(m = 7, n = 3, k = 10) == 52907400

Answer: assert Solution().distanceSum(m = 10, n = 10, k = 50) == 434987024

Answer: assert Solution().distanceSum(m = 6, n = 6, k = 18) == 982764729

Answer: assert Solution().distanceSum(m = 5, n = 10, k = 25) == 329316644

Answer: assert Solution().distanceSum(m = 6, n = 5, k = 20) == 931360310

Answer: assert Solution().distanceSum(m = 7, n = 5, k = 15) == 136117652

Answer: assert Solution().distanceSum(m = 40, n = 40, k = 1200) == 350729036

Answer: assert Solution().distanceSum(m = 15, n = 20, k = 100) == 132852497

Answer: assert Solution().distanceSum(m = 100, n = 100, k = 5) == 666666283

Answer: assert Solution().distanceSum(m = 1000, n = 1, k = 10) == 122738391

Answer: assert Solution().distanceSum(m = 15, n = 15, k = 100) == 268949671

Answer: assert Solution().distanceSum(m = 50, n = 50, k = 500) == 833366694

Answer: assert Solution().distanceSum(m = 75, n = 75, k = 300) == 770041881

Answer: assert Solution().distanceSum(m = 20, n = 5, k = 15) == 541933847

Answer: assert Solution().distanceSum(m = 20, n = 25, k = 100) == 883287723

Answer: assert Solution().distanceSum(m = 1, n = 1000, k = 500) == 349233848

Answer: assert Solution().distanceSum(m = 7, n = 6, k = 20) == 753814653

Answer: assert Solution().distanceSum(m = 30, n = 30, k = 90) == 482659936

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 distanceSum(self, m: int, n: int, k: int) -> int:
    # For each distance d, where 1 < d < m, there are `m - d` ways to choose
    # the two columns that the two pieces are on. For each of the two pieces,
    # there are `n` ways to choose the row that the piece is on.
    # Therefore, the contribution of row differences is
    #   sum(d * (m - d) * n^2), where 1 < d <= m - 1
    # = n^2 * sum(d * m - d^2)
    # = n^2 * (d * m * (m - 1) / 2 - m * (m - 1) * (2m - 1) / 6)
    # = n^2 * (m^3 - m) / 6
    # Similarly, the contribution of column differences is
    #   m^2 * (n^3 - n) / 6
    MOD = 1_000_000_007
    return (n**2 * (m**3 - m) // 6 +
            m**2 * (n**3 - n) // 6) * math.comb(m * n - 2, k - 2) % 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 a positive integer k. Return the sum of the maximum and minimum elements of all subsequences of nums with at most k elements.
Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: nums = [1,2,3], k = 2
Output: 24
Explanation:
The subsequences of nums with at most 2 elements are:

Subsequence
Minimum
Maximum
Sum

[1]
1
1
2

[2]
2
2
4

[3]
3
3
6

[1, 2]
1
2
3

[1, 3]
1
3
4

[2, 3]
2
3
5

Final Total

24

The output would be 24.

Example 2:

Input: nums = [5,0,6], k = 1
Output: 22
Explanation:
For subsequences with exactly 1 element, the minimum and maximum values are the element itself. Therefore, the total is 5 + 5 + 0 + 0 + 6 + 6 = 22.

Example 3:

Input: nums = [1,1,1], k = 2
Output: 12
Explanation:
The subsequences [1, 1] and [1] each appear 3 times. For all of them, the minimum and maximum are both 1. Thus, the total is 12.

Constraints:

1 <= nums.length <= 105
0 <= nums[i] <= 109
1 <= k <= min(70, nums.length)

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

SOLUTION:

Question:
PROBLEM:
You are given an integer array nums and a positive integer k. Return the sum of the maximum and minimum elements of all subsequences of nums with at most k elements.
Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: nums = [1,2,3], k = 2
Output: 24
Explanation:
The subsequences of nums with at most 2 elements are:

Subsequence
Minimum
Maximum
Sum

[1]
1
1
2

[2]
2
2
4

[3]
3
3
6

[1, 2]
1
2
3

[1, 3]
1
3
4

[2, 3]
2
3
5

Final Total

24

The output would be 24.

Example 2:

Input: nums = [5,0,6], k = 1
Output: 22
Explanation:
For subsequences with exactly 1 element, the minimum and maximum values are the element itself. Therefore, the total is 5 + 5 + 0 + 0 + 6 + 6 = 22.

Example 3:

Input: nums = [1,1,1], k = 2
Output: 12
Explanation:
The subsequences [1, 1] and [1] each appear 3 times. For all of them, the minimum and maximum are both 1. Thus, the total is 12.

Constraints:

1 <= nums.length <= 105
0 <= nums[i] <= 109
1 <= k <= min(70, nums.length)

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

SOLUTION:

assert Solution().minMaxSums(nums = [7,8,9,10,11], k = 4) == 540

assert Solution().minMaxSums(nums = [5,0,6], k = 1) == 22

assert Solution().minMaxSums(nums = [4,3,2,1], k = 3) == 70

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

assert Solution().minMaxSums(nums = [10,20,30,40], k = 3) == 700

assert Solution().minMaxSums(nums = [0,0,0,0], k = 4) == 0

assert Solution().minMaxSums(nums = [4,5,6,7], k = 3) == 154

assert Solution().minMaxSums(nums = [1,2,3], k = 2) == 24

assert Solution().minMaxSums(nums = [1,1,1], k = 2) == 12

assert Solution().minMaxSums(nums = [4,7,2,5,3], k = 3) == 212

Answer: assert Solution().minMaxSums(nums = [7,8,9,10,11], k = 4) == 540

Answer: assert Solution().minMaxSums(nums = [5,0,6], k = 1) == 22

Answer: assert Solution().minMaxSums(nums = [1,2,3,4,5,6,7,8,9,10], k = 5) == 7007

Answer: assert Solution().minMaxSums(nums = [4,3,2,1], k = 3) == 70

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

Answer: assert Solution().minMaxSums(nums = [10,20,30,40], k = 3) == 700

Answer: assert Solution().minMaxSums(nums = [10,20,30,40,50], k = 4) == 1800

Answer: assert Solution().minMaxSums(nums = [0,0,0,0], k = 4) == 0

Answer: assert Solution().minMaxSums(nums = [2,4,6,8,10,12,14,16,18,20], k = 7) == 21274

Answer: assert Solution().minMaxSums(nums = [4,5,6,7], k = 3) == 154

Answer: assert Solution().minMaxSums(nums = [1,3,5,7,9,11,13], k = 6) == 1764

Answer: assert Solution().minMaxSums(nums = [10,20,30,40,50], k = 5) == 1860

Answer: assert Solution().minMaxSums(nums = [100,200,300,400,500], k = 2) == 9000

Answer: assert Solution().minMaxSums(nums = [1,2,3], k = 2) == 24

Answer: assert Solution().minMaxSums(nums = [0,10,20,30,40,50], k = 4) == 2800

Answer: assert Solution().minMaxSums(nums = [1,1,1], k = 2) == 12

Answer: assert Solution().minMaxSums(nums = [4,7,2,5,3], k = 3) == 212

Answer: assert Solution().minMaxSums(nums = [9,8,7,6,5,4,3,2,1], k = 3) == 1290

Answer: assert Solution().minMaxSums(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 3) == 14850

Answer: assert Solution().minMaxSums(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 19) == 22020054

Answer: assert Solution().minMaxSums(nums = [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 2, 8, 0, 5, 4, 5, 2, 1], k = 30) == 487717183

Answer: assert Solution().minMaxSums(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25], k = 15) == 768086094

Answer: assert Solution().minMaxSums(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 = 10) == 643282124

Answer: assert Solution().minMaxSums(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], k = 10) == 29289759

Answer: assert Solution().minMaxSums(nums = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], k = 20) == 11534325

Answer: assert Solution().minMaxSums(nums = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 10) == 3699990

Answer: assert Solution().minMaxSums(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 32, 32, 33, 33, 34, 34, 35, 35, 36, 36, 37, 37, 38, 38, 39, 39, 40, 40, 41, 41, 42, 42, 43, 43, 44, 44, 45, 45, 46, 46, 47, 47, 48, 48, 49, 49, 50, 50, 51, 51, 52, 52, 53, 53, 54, 54, 55, 55, 56, 56, 57, 57, 58, 58, 59, 59, 60, 60, 61, 61, 62, 62, 63, 63, 64, 64, 65, 65, 66, 66, 67, 67, 68, 68, 69, 69, 70, 70, 71, 71, 72, 72, 73, 73, 74, 74, 75, 75, 76, 76, 77, 77, 78, 78, 79, 79, 80, 80, 81, 81, 82, 82, 83, 83, 84, 84, 85, 85, 86, 86, 87, 87, 88, 88, 89, 89, 90, 90, 91, 91, 92, 92, 93, 93, 94, 94, 95, 95, 96, 96, 97, 97, 98, 98, 99, 99, 100, 100], k = 35) == 388218872

Answer: assert Solution().minMaxSums(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], k = 15) == 45510223

Answer: assert Solution().minMaxSums(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150], k = 6) == 1591680

Answer: assert Solution().minMaxSums(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], k = 25) == 179358153

Answer: assert Solution().minMaxSums(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 = 1) == 60

Answer: assert Solution().minMaxSums(nums = [1000000000, 999999999, 999999998, 999999997, 999999996], k = 5) == 999999449

Answer: assert Solution().minMaxSums(nums = [50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50, 50], k = 10) == 821501669

Answer: assert Solution().minMaxSums(nums = [1000000000, 999999999, 888888888, 777777777, 666666666, 555555555, 444444444, 333333333, 222222222, 111111111], k = 5) == 999994264

Answer: assert Solution().minMaxSums(nums = [987654321, 876543210, 765432109, 654321098, 543210987, 432109876, 321098765, 210987654, 109876543], k = 7) == 203221875

Answer: assert Solution().minMaxSums(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200], k = 15) == 218899590

Answer: assert Solution().minMaxSums(nums = [50, 25, 75, 100, 200, 150, 175, 125, 225, 275, 300, 10, 20, 30, 40, 60, 80, 90, 110, 130, 140, 160, 180, 190, 210, 230, 240, 260, 280, 290], k = 20) == 780111119

Answer: assert Solution().minMaxSums(nums = [1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000], k = 9) == 197364519

Answer: assert Solution().minMaxSums(nums = [5, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 2, 1], k = 10) == 184956

Answer: assert Solution().minMaxSums(nums = [0, 1000000000, 500000000, 250000000, 750000000, 125000000, 375000000, 625000000, 875000000, 937500000], k = 7) == 937492909

Answer: assert Solution().minMaxSums(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150], k = 10) == 4932160

Answer: assert Solution().minMaxSums(nums = [1, 1000000000, 2, 999999999, 3, 999999998, 4, 999999997, 5, 999999996, 6, 999999995], k = 50) == 999975437

Answer: assert Solution().minMaxSums(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 15) == 11466169

Answer: assert Solution().minMaxSums(nums = [1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000], k = 50) == 197364513

Answer: assert Solution().minMaxSums(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 70) == 22020075

Answer: assert Solution().minMaxSums(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 15) == 21889959

Answer: assert Solution().minMaxSums(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, 5, 5], k = 50) == 655733852

Answer: assert Solution().minMaxSums(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, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150], k = 50) == 577364634

Answer: assert Solution().minMaxSums(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000], k = 30) == 202007486

Answer: assert Solution().minMaxSums(nums = [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, 3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, 6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6], k = 30) == 810728451

Answer: assert Solution().minMaxSums(nums = [999999999, 999999998, 999999997, 999999996, 999999995, 999999994, 999999993, 999999992, 999999991, 999999990], k = 5) == 999984082

Answer: assert Solution().minMaxSums(nums = [1, 3, 2, 4, 6, 5, 8, 7, 10, 9, 12, 11, 14, 13, 15], k = 8) == 365088

Answer: assert Solution().minMaxSums(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 210, 220, 230, 240, 250], k = 20) == 720180244

Answer: assert Solution().minMaxSums(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 5) == 455679

Answer: assert Solution().minMaxSums(nums = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3], k = 30) == 648895755

Answer: assert Solution().minMaxSums(nums = [1000000000, 0, 500000000, 250000000, 750000000, 1, 999999999, 2, 999999998, 3], k = 5) == 999995939

Answer: assert Solution().minMaxSums(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 = 15) == 47319668

Answer: assert Solution().minMaxSums(nums = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10], k = 10) == 6783315

Answer: assert Solution().minMaxSums(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, 2], k = 40) == 389525101

Answer: assert Solution().minMaxSums(nums = [10,10,10,10,10,20,20,20,20,20,30,30,30,30,30,40,40,40,40,40,50,50,50,50,50], k = 20) == 12349286

Answer: assert Solution().minMaxSums(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 = 25) == 285006421

Answer: assert Solution().minMaxSums(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], k = 30) == 737418160

Answer: assert Solution().minMaxSums(nums = [1000000000, 999999999, 999999998, 999999997, 999999996, 999999995, 999999994, 999999993, 999999992, 999999991], k = 10) == 999976478

Answer: assert Solution().minMaxSums(nums = [1, 3, 2, 5, 4, 7, 6, 9, 8, 11, 10, 13, 12, 15, 14, 17, 16, 19, 18, 21, 20, 23, 22, 25, 24], k = 15) == 772295550

Answer: assert Solution().minMaxSums(nums = [1, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100], k = 25) == 210763677

Answer: assert Solution().minMaxSums(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 10) == 6166650

Answer: assert Solution().minMaxSums(nums = [0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1], k = 20) == 50777729

Answer: assert Solution().minMaxSums(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1200, 1300, 1400, 1500, 1600, 1700, 1800, 1900, 2000], k = 4) == 13009500

Answer: assert Solution().minMaxSums(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 = 20) == 208404942

Answer: assert Solution().minMaxSums(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 = 35) == 340232322

Answer: assert Solution().minMaxSums(nums = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130], k = 15) == 578709736

Answer: assert Solution().minMaxSums(nums = [999999999, 999999998, 999999997, 999999996, 999999995, 999999994, 999999993, 999999992, 999999991, 999999990], k = 70) == 999974432

Answer: assert Solution().minMaxSums(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], k = 5) == 79088

Answer: assert Solution().minMaxSums(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 = 10) == 106018202

Answer: assert Solution().minMaxSums(nums = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150, 155, 160, 165, 170, 175, 180, 185, 190, 195, 200, 205, 210, 215, 220, 225, 230, 235, 240, 245, 250, 255, 260, 265, 270, 275, 280, 285, 290, 295, 300, 305, 310, 315, 320, 325, 330, 335, 340, 345, 350, 355, 360, 365, 370, 375, 380, 385, 390, 395, 400], k = 40) == 990544039

Answer: assert Solution().minMaxSums(nums = [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5], k = 6) == 12551

Answer: assert Solution().minMaxSums(nums = [5, 15, 25, 35, 45, 55, 65, 75, 85, 95, 105, 115, 125, 135, 145, 155, 165, 175, 185, 195], k = 12) == 182119000

Answer: assert Solution().minMaxSums(nums = [1, 1000000000, 2, 999999999, 3, 888888888, 4, 777777777, 5, 666666666], k = 5) == 444441332

Answer: assert Solution().minMaxSums(nums = [1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6], k = 4) == 63232

Answer: assert Solution().minMaxSums(nums = [1000000000, 1000000000, 1000000000, 1000000000, 1000000000], k = 50) == 999999573

Answer: assert Solution().minMaxSums(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39], k = 20) == 41943000

Answer: assert Solution().minMaxSums(nums = [1000000000, 1000000000, 1000000000, 1000000000, 1000000000], k = 5) == 999999573

Answer: assert Solution().minMaxSums(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 = 15) == 772295550

Answer: assert Solution().minMaxSums(nums = [100,200,300,400,500,600,700,800,900,1000,1100,1200,1300,1400,1500], k = 10) == 49321600

Answer: assert Solution().minMaxSums(nums = [1000000000, 999999999, 999999998, 999999997, 999999996, 999999995, 999999994, 999999993, 999999992, 999999991], k = 5) == 999985356

Answer: assert Solution().minMaxSums(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], k = 30) == 659413563

Answer: assert Solution().minMaxSums(nums = [9, 8, 7, 6, 5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10], k = 10) == 999383342

Answer: assert Solution().minMaxSums(nums = [5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5], k = 20) == 10485750

Answer: assert Solution().minMaxSums(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 = 20) == 574109592

Answer: assert Solution().minMaxSums(nums = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100], k = 7) == 14487795

Answer: assert Solution().minMaxSums(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 20) == 22020075

Answer: assert Solution().minMaxSums(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], k = 10) == 12949965

Answer: assert Solution().minMaxSums(nums = [5, 4, 3, 2, 1, 0, -1, -2, -3, -4, -5, -6, -7, -8, -9, -10], k = 15) == 999672337

Answer: assert Solution().minMaxSums(nums = [7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2, 1, 7, 6, 5, 4, 3, 2], k = 20) == 8538396

Answer: assert Solution().minMaxSums(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 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], k = 30) == 620077159

Answer: assert Solution().minMaxSums(nums = [7, 3, 9, 1, 5, 8, 4, 6, 2, 0], k = 4) == 3465

Answer: assert Solution().minMaxSums(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, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100], k = 70) == 913257662

Answer: assert Solution().minMaxSums(nums = [1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384], k = 8) == 228752329

Answer: assert Solution().minMaxSums(nums = [1000000000, 0, 500000000, 250000000, 750000000, 125000000, 375000000, 625000000, 875000000, 937500000], k = 5) == 187495282

Answer: assert Solution().minMaxSums(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], k = 35) == 779042774

Answer: assert Solution().minMaxSums(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], k = 10) == 12949965

Answer: assert Solution().minMaxSums(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) == 865112118

Answer: assert Solution().minMaxSums(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200], k = 10) == 129499650

Answer: assert Solution().minMaxSums(nums = [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 9, 7, 9, 3, 2, 3, 8, 4, 6], k = 15) == 10517752

Answer: assert Solution().minMaxSums(nums = [1, 10, 100, 1000, 10000, 100000, 1000000, 10000000, 100000000, 1000000000], k = 7) == 287364606

Answer: assert Solution().minMaxSums(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], k = 50) == 237470545

Answer: assert Solution().minMaxSums(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200], k = 70) == 220200750

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 minMaxSums(self, nums: list[int], k: int) -> int:
    # In a sorted array, nums[i] will be
    #   1. The maximum for subsequences formed by nums[0..i].
    #   2. The minimum for subsequences formed by nums[i..n - 1].
    #
    # The number of times nums[i] is the maximum is the same as the number of
    # times nums[n - 1 - i] is the minimum, due to the symmetry in subsequences
    # derived from the sorted order.
    #
    # To calculate the contribution of nums[i], we need to find the number of
    # ways to select at most (k - 1) elements from the range of indices where
    # nums[i] is the smallest or nums[n - 1 - i] is the largest.
    MOD = 1_000_000_007
    n = len(nums)

    def getComb(n: int, k: int) -> list[list[int]]:
      """C(n, k) = C(n - 1, k) + C(n - 1, k - 1)"""
      comb = [[0] * (k + 1) for _ in range(n + 1)]
      for i in range(n + 1):
        comb[i][0] = 1
      for i in range(1, n + 1):
        for j in range(1, k + 1):
          comb[i][j] = (comb[i - 1][j] + comb[i - 1][j - 1]) % MOD
      return comb

    comb = getComb(n, k - 1)
    ans = 0

    nums.sort()

    # i: available numbers from the left of nums[i] or
    #    available numbers from the right of nums[-1 - i]
    for i in range(n):
      count = 0
      for j in range(k):  # selected numbers
        count = (count + comb[i][j]) % MOD
      ans += nums[i] * count
      ans += nums[-1 - i] * count
      ans %= MOD

    return ans


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

PROBLEM:
You are given an array nums consisting of integers between 1 and 3, and a binary array locked of the same size.
We consider nums sortable if it can be sorted using adjacent swaps, where a swap between two indices i and i + 1 is allowed if nums[i] - nums[i + 1] == 1 and locked[i] == 0.
In one operation, you can unlock any index i by setting locked[i] to 0.
Return the minimum number of operations needed to make nums sortable. If it is not possible to make nums sortable, return -1.

Example 1:

Input: nums = [1,2,1,2,3,2], locked = [1,0,1,1,0,1]
Output: 0
Explanation:
We can sort nums using the following swaps:

swap indices 1 with 2
swap indices 4 with 5

So, there is no need to unlock any index.

Example 2:

Input: nums = [1,2,1,1,3,2,2], locked = [1,0,1,1,0,1,0]
Output: 2
Explanation:
If we unlock indices 2 and 5, we can sort nums using the following swaps:

swap indices 1 with 2
swap indices 2 with 3
swap indices 4 with 5
swap indices 5 with 6

Example 3:

Input: nums = [1,2,1,2,3,2,1], locked = [0,0,0,0,0,0,0]
Output: -1
Explanation:
Even if all indices are unlocked, it can be shown that nums is not sortable.

Constraints:

1 <= nums.length <= 105
1 <= nums[i] <= 3
locked.length == nums.length
0 <= locked[i] <= 1

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

SOLUTION:

Question:
PROBLEM:
You are given an array nums consisting of integers between 1 and 3, and a binary array locked of the same size.
We consider nums sortable if it can be sorted using adjacent swaps, where a swap between two indices i and i + 1 is allowed if nums[i] - nums[i + 1] == 1 and locked[i] == 0.
In one operation, you can unlock any index i by setting locked[i] to 0.
Return the minimum number of operations needed to make nums sortable. If it is not possible to make nums sortable, return -1.

Example 1:

Input: nums = [1,2,1,2,3,2], locked = [1,0,1,1,0,1]
Output: 0
Explanation:
We can sort nums using the following swaps:

swap indices 1 with 2
swap indices 4 with 5

So, there is no need to unlock any index.

Example 2:

Input: nums = [1,2,1,1,3,2,2], locked = [1,0,1,1,0,1,0]
Output: 2
Explanation:
If we unlock indices 2 and 5, we can sort nums using the following swaps:

swap indices 1 with 2
swap indices 2 with 3
swap indices 4 with 5
swap indices 5 with 6

Example 3:

Input: nums = [1,2,1,2,3,2,1], locked = [0,0,0,0,0,0,0]
Output: -1
Explanation:
Even if all indices are unlocked, it can be shown that nums is not sortable.

Constraints:

1 <= nums.length <= 105
1 <= nums[i] <= 3
locked.length == nums.length
0 <= locked[i] <= 1

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

SOLUTION:

assert Solution().minUnlockedIndices(nums = [1,2,3,3,2,1], locked = [0,1,0,1,0,1]) == -1

assert Solution().minUnlockedIndices(nums = [3,2,1,1,2,1], locked = [1,1,1,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [2,1,2,1,3,1], locked = [0,0,0,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [2,1,1,2,3,3], locked = [1,0,1,0,1,0]) == 1

assert Solution().minUnlockedIndices(nums = [1,2,1,2,3,2,1], locked = [0,0,0,0,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [1,3,2,1,3,2], locked = [0,0,0,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [3,3,3,3,3,3], locked = [0,0,0,0,0,0]) == 0

assert Solution().minUnlockedIndices(nums = [2,2,2,2,2,2], locked = [1,1,1,1,1,1]) == 0

assert Solution().minUnlockedIndices(nums = [1,1,2,2,3,3], locked = [1,1,1,1,1,1]) == 0

assert Solution().minUnlockedIndices(nums = [2,2,2,2,2], locked = [0,0,0,0,0]) == 0

assert Solution().minUnlockedIndices(nums = [1,1,2,2,3,3], locked = [0,0,0,0,0,0]) == 0

assert Solution().minUnlockedIndices(nums = [1,2,3,1,2,3], locked = [0,1,0,1,0,1]) == -1

assert Solution().minUnlockedIndices(nums = [3,2,1,2,1,1], locked = [1,1,1,1,1,1]) == -1

assert Solution().minUnlockedIndices(nums = [1,2,3,1,2,3], locked = [0,0,0,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [1,2,1,2,3,2], locked = [1,0,1,1,0,1]) == 0

assert Solution().minUnlockedIndices(nums = [3,3,3,3,3], locked = [1,1,1,1,1]) == 0

assert Solution().minUnlockedIndices(nums = [3,2,1,2,1,1], locked = [0,1,0,1,0,1]) == -1

assert Solution().minUnlockedIndices(nums = [3,3,2,2,1,1], locked = [1,1,1,1,1,1]) == -1

assert Solution().minUnlockedIndices(nums = [3,1,2,3,1,2], locked = [1,1,1,1,1,1]) == -1

assert Solution().minUnlockedIndices(nums = [1,1,1,1,1,1], locked = [0,0,0,0,0,0]) == 0

assert Solution().minUnlockedIndices(nums = [3,2,1,3,2,1], locked = [1,1,1,1,1,1]) == -1

assert Solution().minUnlockedIndices(nums = [3,2,1], locked = [0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [1,3,2,1,2,3], locked = [1,1,1,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [1,2,1,1,3,2,2], locked = [1,0,1,1,0,1,0]) == 2

assert Solution().minUnlockedIndices(nums = [2,3,1,3,1,2], locked = [0,0,0,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [1,1,2,2,3,3], locked = [0,1,0,1,0,1]) == 0

assert Solution().minUnlockedIndices(nums = [1,3,2,3,2,1], locked = [1,1,1,1,1,1]) == -1

assert Solution().minUnlockedIndices(nums = [1,2,3,2,1,2], locked = [1,1,1,1,1,1]) == -1

assert Solution().minUnlockedIndices(nums = [1,3,2], locked = [1,1,1]) == 1

assert Solution().minUnlockedIndices(nums = [3,2,3,1,2,1], locked = [1,0,1,0,1,0]) == -1

assert Solution().minUnlockedIndices(nums = [1,1,3,2,2,3], locked = [1,1,1,1,0,0]) == 2

assert Solution().minUnlockedIndices(nums = [3,2,1,3,1,2,1], locked = [1,1,1,1,1,1,1]) == -1

assert Solution().minUnlockedIndices(nums = [1,3,2,3,2,1], locked = [0,1,0,1,0,1]) == -1

assert Solution().minUnlockedIndices(nums = [2,3,1,2,3,1,2], locked = [1,0,1,0,1,0,1]) == -1

assert Solution().minUnlockedIndices(nums = [1,3,2,1,3,2], locked = [1,1,1,0,0,0]) == -1

assert Solution().minUnlockedIndices(nums = [3,1,3,2,3,1,2], locked = [1,1,0,1,0,1,0]) == -1

assert Solution().minUnlockedIndices(nums = [1,3,2,1,3,2,3], locked = [0,1,0,1,0,1,0]) == -1

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [1,1,1,2,2,2,3,3,3], locked = [1,1,1,1,1,1,1,1,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [2,1,1,2,3,3], locked = [1,0,1,0,1,0]) == 1

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

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

Answer: assert Solution().minUnlockedIndices(nums = [3,3,3,3,3,3], locked = [0,0,0,0,0,0]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [2,2,2,2,2,2], locked = [1,1,1,1,1,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [1,1,2,2,3,3], locked = [1,1,1,1,1,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [2,2,2,2,2], locked = [0,0,0,0,0]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [1,1,2,2,3,3], locked = [0,0,0,0,0,0]) == 0

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

Answer: assert Solution().minUnlockedIndices(nums = [3,2,1,2,1,1], locked = [1,1,1,1,1,1]) == -1

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

Answer: assert Solution().minUnlockedIndices(nums = [1,2,1,2,3,2], locked = [1,0,1,1,0,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [3,3,3,3,3], locked = [1,1,1,1,1]) == 0

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

Answer: assert Solution().minUnlockedIndices(nums = [3,3,2,2,1,1], locked = [1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [3,1,2,3,1,2], locked = [1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [1,1,1,1,1,1], locked = [0,0,0,0,0,0]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [3,2,1,3,2,1], locked = [1,1,1,1,1,1]) == -1

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

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

Answer: assert Solution().minUnlockedIndices(nums = [1,2,1,1,3,2,2], locked = [1,0,1,1,0,1,0]) == 2

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

Answer: assert Solution().minUnlockedIndices(nums = [1,1,2,2,3,3], locked = [0,1,0,1,0,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [1,3,2,3,2,1], locked = [1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [1,2,3,2,1,2], locked = [1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [1,3,2], locked = [1,1,1]) == 1

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [3,2,1,3,2,1,3,2,1,3,2,1], locked = [1,1,1,1,1,1,1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [3,2,1,2,3,1,2,3,1,2,1,3,2,1,3,2,1,3,2,1], locked = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]) == -1

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

Answer: assert Solution().minUnlockedIndices(nums = [3,3,3,2,2,2,1,1,1], locked = [1,1,1,1,1,1,1,1,1]) == -1

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

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

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

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

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [1,3,1,3,1,3,1,3,1,3], locked = [0,1,0,1,0,1,0,1,0,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [2,3,2,3,2,3,2,3], locked = [1,0,1,0,1,0,1,0]) == 2

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

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

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [1,1,1,1,2,2,2,2,3,3,3,3], locked = [0,1,0,1,0,1,0,1,0,1,0,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [1,2,3,1,2,3,1,2,3,1,2,3], locked = [1,1,1,1,1,1,1,1,1,1,1,1]) == -1

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

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

Answer: assert Solution().minUnlockedIndices(nums = [1,2,1,2,1,2,1,2,1,2,1,2], locked = [0,0,0,0,0,0,0,0,0,0,0,0]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [1,1,3,2,2,3], locked = [1,1,1,1,0,0]) == 2

Answer: assert Solution().minUnlockedIndices(nums = [3,2,3,2,3,2,3,2,3,2,3,2], locked = [1,1,1,1,1,1,1,1,1,1,1,1]) == 11

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

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

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

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [1,1,1,2,2,2,3,3,3], locked = [0,0,0,0,0,0,0,0,0]) == 0

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

Answer: assert Solution().minUnlockedIndices(nums = [1,3,1,3,1,3,1,3,1,3,1,3,1,3,1], locked = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]) == -1

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

Answer: assert Solution().minUnlockedIndices(nums = [3,1,2,1,2,3,1,2,3], locked = [1,1,1,1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [2,1,2,1,2,1,2,1,2,1,2,1], locked = [1,0,1,0,1,0,1,0,1,0,1,0]) == 6

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

Answer: assert Solution().minUnlockedIndices(nums = [3,2,3,2,3,2,3,2,3,2,3,2,3], locked = [1,1,1,1,1,1,1,1,1,1,1,1,1]) == 11

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

Answer: assert Solution().minUnlockedIndices(nums = [1,1,1,2,2,2,3,3,3], locked = [1,1,1,0,0,0,1,1,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [2,1,3,2,1,3,2,1,3,2], locked = [1,1,1,1,1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], locked = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]) == 0

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

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

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

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

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

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

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [3,2,1,3,1,2,1], locked = [1,1,1,1,1,1,1]) == -1

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

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

Answer: assert Solution().minUnlockedIndices(nums = [3,2,1,2,3,1,2,3,1,2,3], locked = [1,1,1,1,1,1,1,1,1,1,1]) == -1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [2,1,2,1,2,1,2,1,2,1,2,1,2], locked = [0,0,0,0,0,0,0,0,0,0,0,0,0]) == 0

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [2,1,3,2,1,3,2,1,3], locked = [1,1,1,1,1,1,1,1,1]) == -1

Answer: assert Solution().minUnlockedIndices(nums = [1,1,1,2,2,2,3,3,3], locked = [0,1,1,0,1,1,0,1,1]) == 0

Answer: assert Solution().minUnlockedIndices(nums = [1,3,1,3,1,3,1,3,1], locked = [1,1,1,1,1,1,1,1,1]) == -1

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

Answer: assert Solution().minUnlockedIndices(nums = [1,2,1,3,2,1,3,2,1,3,2,1], locked = [1,1,1,1,1,1,1,1,1,1,1,1]) == -1

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer: assert Solution().minUnlockedIndices(nums = [1,2,3,1,2,3,1,2,3], locked = [1,0,1,1,0,1,1,0,1]) == -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 minUnlockedIndices(self, nums: List[int], locked: List[int]) -> int:
        n = len(nums)
        first2 = first3 = n
        last1 = last2 = -1
        for i, x in enumerate(nums):
            if x == 1:
                last1 = i
            elif x == 2:
                first2 = min(first2, i)
                last2 = i
            else:
                first3 = min(first3, i)
        if first3 < last1:
            return -1
        return sum(
            st and (first2 <= i < last1 or first3 <= i < last2)
            for i, st in enumerate(locked)
        )