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 original of length n and a 2D array bounds of length n x 2, where bounds[i] = [ui, vi].
You need to find the number of possible arrays copy of length n such that:

(copy[i] - copy[i - 1]) == (original[i] - original[i - 1]) for 1 <= i <= n - 1.
ui <= copy[i] <= vi for 0 <= i <= n - 1.

Return the number of such arrays.

Example 1:

Input: original = [1,2,3,4], bounds = [[1,2],[2,3],[3,4],[4,5]]
Output: 2
Explanation:
The possible arrays are:

[1, 2, 3, 4]
[2, 3, 4, 5]

Example 2:

Input: original = [1,2,3,4], bounds = [[1,10],[2,9],[3,8],[4,7]]
Output: 4
Explanation:
The possible arrays are:

[1, 2, 3, 4]
[2, 3, 4, 5]
[3, 4, 5, 6]
[4, 5, 6, 7]

Example 3:

Input: original = [1,2,1,2], bounds = [[1,1],[2,3],[3,3],[2,3]]
Output: 0
Explanation:
No array is possible.

Constraints:

2 <= n == original.length <= 105
1 <= original[i] <= 109
bounds.length == n
bounds[i].length == 2
1 <= bounds[i][0] <= bounds[i][1] <= 109

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

SOLUTION:

Question:
PROBLEM:
You are given an array original of length n and a 2D array bounds of length n x 2, where bounds[i] = [ui, vi].
You need to find the number of possible arrays copy of length n such that:

(copy[i] - copy[i - 1]) == (original[i] - original[i - 1]) for 1 <= i <= n - 1.
ui <= copy[i] <= vi for 0 <= i <= n - 1.

Return the number of such arrays.

Example 1:

Input: original = [1,2,3,4], bounds = [[1,2],[2,3],[3,4],[4,5]]
Output: 2
Explanation:
The possible arrays are:

[1, 2, 3, 4]
[2, 3, 4, 5]

Example 2:

Input: original = [1,2,3,4], bounds = [[1,10],[2,9],[3,8],[4,7]]
Output: 4
Explanation:
The possible arrays are:

[1, 2, 3, 4]
[2, 3, 4, 5]
[3, 4, 5, 6]
[4, 5, 6, 7]

Example 3:

Input: original = [1,2,1,2], bounds = [[1,1],[2,3],[3,3],[2,3]]
Output: 0
Explanation:
No array is possible.

Constraints:

2 <= n == original.length <= 105
1 <= original[i] <= 109
bounds.length == n
bounds[i].length == 2
1 <= bounds[i][0] <= bounds[i][1] <= 109

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

SOLUTION:

assert Solution().countArrays(original = [3,2,1], bounds = [[1,3],[1,3],[1,3]]) == 1

assert Solution().countArrays(original = [5,3,1], bounds = [[1,5],[2,5],[3,5]]) == 0

Answer: assert Solution().countArrays(original = [1,2,3,4], bounds = [[1,10],[2,9],[3,8],[4,7]]) == 4

Answer: assert Solution().countArrays(original = [3,5,7,9], bounds = [[2,4],[4,6],[6,8],[8,10]]) == 3

Answer: assert Solution().countArrays(original = [10,11,12,13], bounds = [[8,10],[9,11],[10,12],[11,13]]) == 3

Answer: assert Solution().countArrays(original = [5,7,9,11], bounds = [[4,6],[6,8],[8,10],[10,12]]) == 3

Answer: assert Solution().countArrays(original = [2,2,2,2], bounds = [[1,3],[1,3],[1,3],[1,3]]) == 3

Answer: assert Solution().countArrays(original = [3,2,1], bounds = [[1,3],[1,3],[1,3]]) == 1

Answer: assert Solution().countArrays(original = [5,6,7,8], bounds = [[5,5],[6,6],[7,7],[8,8]]) == 1

Answer: assert Solution().countArrays(original = [5,6,7,8], bounds = [[3,5],[4,6],[5,7],[6,8]]) == 3

Answer: assert Solution().countArrays(original = [1,3,5,7], bounds = [[1,2],[3,4],[5,6],[7,8]]) == 2

Answer: assert Solution().countArrays(original = [1,3,2,4], bounds = [[1,3],[2,4],[1,3],[2,4]]) == 1

Answer: assert Solution().countArrays(original = [3,2,1,2], bounds = [[1,3],[2,4],[1,3],[2,4]]) == 1

Answer: assert Solution().countArrays(original = [1,5,9,13], bounds = [[1,4],[5,8],[9,12],[13,16]]) == 4

Answer: assert Solution().countArrays(original = [1,2,3,4], bounds = [[1,2],[2,3],[3,4],[4,5]]) == 2

Answer: assert Solution().countArrays(original = [3,2,1,4], bounds = [[1,5],[1,5],[1,5],[1,5]]) == 2

Answer: assert Solution().countArrays(original = [4,3,2,1], bounds = [[2,5],[2,5],[2,5],[2,5]]) == 1

Answer: assert Solution().countArrays(original = [5,5,5,5], bounds = [[1,5],[1,5],[1,5],[1,5]]) == 5

Answer: assert Solution().countArrays(original = [1,2,1,2], bounds = [[1,1],[2,3],[3,3],[2,3]]) == 0

Answer: assert Solution().countArrays(original = [100, 200, 150, 250, 300], bounds = [[50, 150], [100, 250], [100, 200], [150, 300], [200, 350]]) == 101

Answer: assert Solution().countArrays(original = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3], bounds = [[1,3],[1,3],[1,3],[1,3],[1,3],[1,3],[1,3],[1,3],[1,3],[1,3]]) == 3

Answer: assert Solution().countArrays(original = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5], bounds = [[1, 2], [1, 2], [2, 3], [2, 3], [3, 4], [3, 4], [4, 5], [4, 5], [5, 6], [5, 6]]) == 2

Answer: assert Solution().countArrays(original = [10, 20, 30, 40, 50], bounds = [[5,15],[15,25],[25,35],[35,45],[45,55]]) == 11

Answer: assert Solution().countArrays(original = [10,9,8,7,6], bounds = [[5,10],[5,10],[5,10],[5,10],[5,10]]) == 2

Answer: assert Solution().countArrays(original = [5,7,9,11,13], bounds = [[3,6],[5,8],[7,10],[9,12],[11,14]]) == 4

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

Answer: assert Solution().countArrays(original = [9, 7, 5, 3, 1], bounds = [[5, 10], [3, 8], [1, 6], [2, 7], [1, 4]]) == 2

Answer: assert Solution().countArrays(original = [1,2,3,4,5], bounds = [[0,3],[1,4],[2,5],[3,6],[4,7]]) == 4

Answer: assert Solution().countArrays(original = [10, 20, 30, 40], bounds = [[5, 15], [15, 25], [25, 35], [35, 45]]) == 11

Answer: assert Solution().countArrays(original = [1,3,6,10,15], bounds = [[1,10],[2,11],[5,14],[9,18],[14,23]]) == 9

Answer: assert Solution().countArrays(original = [10, 20, 10, 20, 10], bounds = [[5, 15], [15, 25], [5, 15], [15, 25], [5, 15]]) == 11

Answer: assert Solution().countArrays(original = [1,2,2,1,2], bounds = [[1,3],[2,3],[2,3],[1,3],[2,3]]) == 2

Answer: assert Solution().countArrays(original = [1,4,9,16,25], bounds = [[1,10],[4,15],[9,20],[16,25],[25,30]]) == 6

Answer: assert Solution().countArrays(original = [5, 6, 7, 8, 9], bounds = [[3, 5], [4, 6], [5, 7], [6, 8], [7, 9]]) == 3

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

Answer: assert Solution().countArrays(original = [10,9,8,7,6], bounds = [[5,10],[4,9],[3,8],[2,7],[1,6]]) == 6

Answer: assert Solution().countArrays(original = [1, 2, 2, 2, 3, 4, 4, 4, 5, 6], bounds = [[1, 1], [2, 2], [2, 2], [2, 2], [3, 3], [4, 4], [4, 4], [4, 4], [5, 5], [6, 6]]) == 1

Answer: assert Solution().countArrays(original = [1,3,5,7,9,11], bounds = [[1,2],[2,4],[4,6],[6,8],[8,10],[10,12]]) == 2

Answer: assert Solution().countArrays(original = [10,20,30,40], bounds = [[5,15],[15,25],[25,35],[35,45]]) == 11

Answer: assert Solution().countArrays(original = [10,15,20,25,30], bounds = [[8,12],[13,17],[18,22],[23,27],[28,32]]) == 5

Answer: assert Solution().countArrays(original = [1,1,1,1,1], bounds = [[1,1],[1,1],[1,1],[1,1],[1,1]]) == 1

Answer: assert Solution().countArrays(original = [10, 20, 30, 40], bounds = [[5, 15],[15, 25],[25, 35],[35, 45]]) == 11

Answer: assert Solution().countArrays(original = [3,6,12,24,48], bounds = [[2,4],[5,7],[11,13],[23,25],[47,49]]) == 3

Answer: assert Solution().countArrays(original = [5,4,3,2,1], bounds = [[2,7],[1,6],[0,5],[0,4],[0,3]]) == 4

Answer: assert Solution().countArrays(original = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], bounds = [[1, 3], [2, 4], [3, 5], [4, 6], [5, 7], [6, 8], [7, 9], [8, 10], [9, 11], [10, 12]]) == 3

Answer: assert Solution().countArrays(original = [10,15,20,25,30], bounds = [[5,10],[10,15],[15,20],[20,25],[25,30]]) == 6

Answer: assert Solution().countArrays(original = [5,4,3,2,1], bounds = [[1,10],[1,10],[1,10],[1,10],[1,10]]) == 6

Answer: assert Solution().countArrays(original = [100, 99, 98, 97, 96], bounds = [[95, 105], [94, 104], [93, 103], [92, 102], [91, 101]]) == 11

Answer: assert Solution().countArrays(original = [1,1,1,1,1], bounds = [[1,2],[1,2],[1,2],[1,2],[1,2]]) == 2

Answer: assert Solution().countArrays(original = [1000000000, 1000000000, 1000000000, 1000000000], bounds = [[1, 1000000000], [1, 1000000000], [1, 1000000000], [1, 1000000000]]) == 1000000000

Answer: assert Solution().countArrays(original = [1,2,4,8,16], bounds = [[1,3],[2,5],[4,7],[8,11],[16,19]]) == 3

Answer: assert Solution().countArrays(original = [1,2,3,2,1], bounds = [[1,3],[2,4],[3,5],[2,4],[1,3]]) == 3

Answer: assert Solution().countArrays(original = [1,4,7,10], bounds = [[1,5],[3,8],[6,11],[9,14]]) == 5

Answer: assert Solution().countArrays(original = [1,10,100,1000,10000], bounds = [[1,5],[5,50],[50,500],[500,5000],[5000,50000]]) == 5

Answer: assert Solution().countArrays(original = [1,1,2,2], bounds = [[1,2],[1,2],[1,2],[1,2]]) == 1

Answer: assert Solution().countArrays(original = [1, 2, 3, 2, 1], bounds = [[1, 2], [2, 3], [3, 4], [2, 3], [1, 2]]) == 2

Answer: assert Solution().countArrays(original = [1, 3, 5, 7], bounds = [[1,10],[3,12],[5,14],[7,16]]) == 10

Answer: assert Solution().countArrays(original = [1,1,1,1,1,1], bounds = [[1,1],[1,1],[1,1],[1,1],[1,1],[1,1]]) == 1

Answer: assert Solution().countArrays(original = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], bounds = [[1,20],[10,30],[20,40],[30,50],[40,60],[50,70],[60,80],[70,90],[80,100],[90,110]]) == 20

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

Answer: assert Solution().countArrays(original = [9, 8, 9, 8, 9, 8, 9, 8, 9, 8], bounds = [[7, 11], [7, 11], [7, 11], [7, 11], [7, 11], [7, 11], [7, 11], [7, 11], [7, 11], [7, 11]]) == 4

Answer: assert Solution().countArrays(original = [1, 2, 4, 8, 16], bounds = [[1,32],[2,32],[4,32],[8,32],[16,32]]) == 17

Answer: assert Solution().countArrays(original = [1, 3, 5, 7, 9], bounds = [[1, 2], [3, 4], [5, 6], [7, 8], [9, 10]]) == 2

Answer: assert Solution().countArrays(original = [1,2,3,4,5], bounds = [[1,1],[2,2],[3,3],[4,4],[5,5]]) == 1

Answer: assert Solution().countArrays(original = [10, 15, 20, 25, 30], bounds = [[5, 12], [10, 17], [15, 22], [20, 27], [25, 32]]) == 8

Answer: assert Solution().countArrays(original = [5,4,3,2], bounds = [[2,6],[2,6],[2,6],[2,6]]) == 2

Answer: assert Solution().countArrays(original = [1,2,3,2,1], bounds = [[1,2],[2,3],[3,4],[2,3],[1,2]]) == 2

Answer: assert Solution().countArrays(original = [5,1,5,1,5], bounds = [[3,7],[1,5],[3,7],[1,5],[3,7]]) == 3

Answer: assert Solution().countArrays(original = [1, 2, 1, 2, 1], bounds = [[1, 2], [2, 3], [1, 2], [2, 3], [1, 2]]) == 2

Answer: assert Solution().countArrays(original = [10, 5, 0, -5, -10], bounds = [[8, 12], [3, 7], [-2, 2], [-7, -3], [-12, -8]]) == 5

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

Answer: assert Solution().countArrays(original = [5, 7, 9, 11, 13], bounds = [[4, 6], [6, 8], [8, 10], [10, 12], [12, 14]]) == 3

Answer: assert Solution().countArrays(original = [1, 2, 3, 4, 5], bounds = [[1, 10], [2, 11], [3, 12], [4, 13], [5, 14]]) == 10

Answer: assert Solution().countArrays(original = [2,4,6,8,10], bounds = [[1,3],[3,5],[5,7],[7,9],[9,11]]) == 3

Answer: assert Solution().countArrays(original = [1, 1, 1, 1, 1, 1, 1], bounds = [[1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1], [1, 1]]) == 1

Answer: assert Solution().countArrays(original = [1, 3, 5, 7, 9], bounds = [[1,2],[2,4],[3,5],[5,7],[6,8]]) == 0

Answer: assert Solution().countArrays(original = [1,10,1,10,1], bounds = [[1,5],[6,15],[1,5],[6,15],[1,5]]) == 5

Answer: assert Solution().countArrays(original = [100,200,100,200], bounds = [[50,150],[150,250],[50,150],[150,250]]) == 101

Answer: assert Solution().countArrays(original = [1,2,1,2,1,2,1,2], bounds = [[1,1],[2,3],[1,1],[2,3],[1,1],[2,3],[1,1],[2,3]]) == 1

Answer: assert Solution().countArrays(original = [1, 1, 1, 1, 1], bounds = [[1,1],[2,2],[3,3],[4,4],[5,5]]) == 0

Answer: assert Solution().countArrays(original = [1,2,1,2,1], bounds = [[1,3],[2,4],[1,3],[2,4],[1,3]]) == 3

Answer: assert Solution().countArrays(original = [5,3,1], bounds = [[1,5],[2,5],[3,5]]) == 0

Answer: assert Solution().countArrays(original = [1,4,9,16,25,36,49,64,81,100], bounds = [[1,20],[3,30],[7,40],[15,50],[24,60],[35,70],[46,80],[57,90],[68,100],[79,110]]) == 11

Answer: assert Solution().countArrays(original = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], bounds = [[5, 15], [15, 25], [25, 35], [35, 45], [45, 55], [55, 65], [65, 75], [75, 85], [85, 95], [95, 105]]) == 11

Answer: assert Solution().countArrays(original = [1,3,2,4], bounds = [[1,3],[2,4],[1,3],[3,5]]) == 2

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

Answer: assert Solution().countArrays(original = [1,3,6,10,15], bounds = [[1,3],[2,6],[5,10],[9,15],[14,20]]) == 3

Answer: assert Solution().countArrays(original = [1,3,6,10,15], bounds = [[1,2],[3,5],[6,8],[9,12],[13,15]]) == 1

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

Answer: assert Solution().countArrays(original = [7, 5, 3, 1, -1], bounds = [[0, 10], [0, 10], [0, 10], [0, 10], [0, 10]]) == 3

Answer: assert Solution().countArrays(original = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], bounds = [[1, 2], [2, 3], [3, 4], [4, 5], [5, 6], [6, 7], [7, 8], [8, 9], [9, 10], [10, 11]]) == 2

Answer: assert Solution().countArrays(original = [1,3,5,7], bounds = [[1,5],[3,7],[5,9],[7,11]]) == 5

Answer: assert Solution().countArrays(original = [1,3,5,7,9], bounds = [[1,2],[2,4],[4,6],[6,8],[8,10]]) == 2

Answer: assert Solution().countArrays(original = [1, 3, 2, 4, 3], bounds = [[1, 3], [2, 4], [1, 3], [2, 4], [1, 3]]) == 1

Answer: assert Solution().countArrays(original = [10,20,30,40,50,60,70,80,90,100], bounds = [[5,15],[15,25],[25,35],[35,45],[45,55],[55,65],[65,75],[75,85],[85,95],[95,105]]) == 11

Answer: assert Solution().countArrays(original = [1,2,3,4,5,6,7,8,9,10], bounds = [[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10],[10,11]]) == 2

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

Answer: assert Solution().countArrays(original = [10, 9, 8, 7, 6], bounds = [[5, 15], [5, 15], [5, 15], [5, 15], [5, 15]]) == 7

Answer: assert Solution().countArrays(original = [1,1,1,1,1,1,1,1,1,1], bounds = [[1,1],[1,1],[1,1],[1,1],[1,1],[1,1],[1,1],[1,1],[1,1],[1,1]]) == 1

Answer: assert Solution().countArrays(original = [5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5], bounds = [[1,15],[2,16],[3,17],[4,18],[5,19],[6,20],[5,19],[4,18],[3,17],[2,16],[1,15]]) == 15

Answer: assert Solution().countArrays(original = [5, 5, 5, 5, 5], bounds = [[1,10],[1,10],[1,10],[1,10],[1,10]]) == 10

Answer: assert Solution().countArrays(original = [100,90,80,70,60], bounds = [[80,120],[70,110],[60,100],[50,90],[40,80]]) == 41

Answer: assert Solution().countArrays(original = [5, 4, 3, 2, 1], bounds = [[1,10],[1,10],[1,10],[1,10],[1,10]]) == 6

Answer: assert Solution().countArrays(original = [10,20,30,40,50], bounds = [[5,15],[15,25],[25,35],[35,45],[45,55]]) == 11

Answer: assert Solution().countArrays(original = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], bounds = [[1, 10], [1, 10], [1, 10], [1, 10], [1, 10], [1, 10], [1, 10], [1, 10], [1, 10], [1, 10]]) == 10

Answer: assert Solution().countArrays(original = [1, 2, 1, 2, 1], bounds = [[1,3],[2,4],[1,3],[2,4],[1,3]]) == 3

Answer: assert Solution().countArrays(original = [100,200,300,200,100], bounds = [[50,150],[150,250],[250,350],[150,250],[50,150]]) == 101

Answer: assert Solution().countArrays(original = [1, 10, 20, 30, 40, 50, 60, 70, 80, 90], bounds = [[1,11],[10,21],[20,31],[30,41],[40,51],[50,61],[60,71],[70,81],[80,91],[90,101]]) == 11

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 countArrays(self, original: list[int], bounds: list[list[int]]) -> int:
    mn, mx = bounds[0]

    for i in range(1, len(original)):
      diff = original[i] - original[i - 1]
      mn = max(mn + diff, bounds[i][0])
      mx = min(mx + diff, bounds[i][1])

    return max(0, mx - mn + 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:
You are given an integer array nums. Your task is to remove all elements from the array by performing one of the following operations at each step until nums is empty:

Choose any two elements from the first three elements of nums and remove them. The cost of this operation is the maximum of the two elements removed.
If fewer than three elements remain in nums, remove all the remaining elements in a single operation. The cost of this operation is the maximum of the remaining elements.

Return the minimum cost required to remove all the elements.

Example 1:

Input: nums = [6,2,8,4]
Output: 12
Explanation:
Initially, nums = [6, 2, 8, 4].

In the first operation, remove nums[0] = 6 and nums[2] = 8 with a cost of max(6, 8) = 8. Now, nums = [2, 4].
In the second operation, remove the remaining elements with a cost of max(2, 4) = 4.

The cost to remove all elements is 8 + 4 = 12. This is the minimum cost to remove all elements in nums. Hence, the output is 12.

Example 2:

Input: nums = [2,1,3,3]
Output: 5
Explanation:
Initially, nums = [2, 1, 3, 3].

In the first operation, remove nums[0] = 2 and nums[1] = 1 with a cost of max(2, 1) = 2. Now, nums = [3, 3].
In the second operation remove the remaining elements with a cost of max(3, 3) = 3.

The cost to remove all elements is 2 + 3 = 5. This is the minimum cost to remove all elements in nums. Hence, the output is 5.

Constraints:

1 <= nums.length <= 1000
1 <= nums[i] <= 106

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, nums: List[int]) -> int:
        # Add your code here
```

SOLUTION:

Question:
PROBLEM:
You are given an integer array nums. Your task is to remove all elements from the array by performing one of the following operations at each step until nums is empty:

Choose any two elements from the first three elements of nums and remove them. The cost of this operation is the maximum of the two elements removed.
If fewer than three elements remain in nums, remove all the remaining elements in a single operation. The cost of this operation is the maximum of the remaining elements.

Return the minimum cost required to remove all the elements.

Example 1:

Input: nums = [6,2,8,4]
Output: 12
Explanation:
Initially, nums = [6, 2, 8, 4].

In the first operation, remove nums[0] = 6 and nums[2] = 8 with a cost of max(6, 8) = 8. Now, nums = [2, 4].
In the second operation, remove the remaining elements with a cost of max(2, 4) = 4.

The cost to remove all elements is 8 + 4 = 12. This is the minimum cost to remove all elements in nums. Hence, the output is 12.

Example 2:

Input: nums = [2,1,3,3]
Output: 5
Explanation:
Initially, nums = [2, 1, 3, 3].

In the first operation, remove nums[0] = 2 and nums[1] = 1 with a cost of max(2, 1) = 2. Now, nums = [3, 3].
In the second operation remove the remaining elements with a cost of max(3, 3) = 3.

The cost to remove all elements is 2 + 3 = 5. This is the minimum cost to remove all elements in nums. Hence, the output is 5.

Constraints:

1 <= nums.length <= 1000
1 <= nums[i] <= 106

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, nums: List[int]) -> int:
        # Add your code here
```

SOLUTION:

assert Solution().minCost(nums = [10,10,10,10]) == 20

assert Solution().minCost(nums = [10,20,30]) == 40

assert Solution().minCost(nums = [999999,1]) == 999999

assert Solution().minCost(nums = [5,3,7,9,1,4]) == 18

assert Solution().minCost(nums = [1]) == 1

assert Solution().minCost(nums = [5,5,5,5,5]) == 15

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

assert Solution().minCost(nums = [6,2,8,4]) == 12

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

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

assert Solution().minCost(nums = [5,5,5,5,5,5]) == 15

assert Solution().minCost(nums = [1,2]) == 2

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

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

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

assert Solution().minCost(nums = [1000000]) == 1000000

assert Solution().minCost(nums = [9, 1, 5, 7, 3, 8, 6]) == 25

assert Solution().minCost(nums = [5, 3, 8, 6, 7, 2, 4]) == 21

assert Solution().minCost(nums = [5,3,8,1,9,2,7]) == 22

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

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

Answer: assert Solution().minCost(nums = [999999,1]) == 999999

Answer: assert Solution().minCost(nums = [5,3,7,9,1,4]) == 18

Answer: assert Solution().minCost(nums = [9,8,7,6,5,4,3,2,1]) == 25

Answer: assert Solution().minCost(nums = [1]) == 1

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

Answer: assert Solution().minCost(nums = [10,20,30,40,50]) == 90

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

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

Answer: assert Solution().minCost(nums = [999999, 999998, 999997, 999996, 999995]) == 2999991

Answer: assert Solution().minCost(nums = [6,2,8,4]) == 12

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

Answer: assert Solution().minCost(nums = [1000000,1,2,3]) == 1000002

Answer: assert Solution().minCost(nums = [1000000,1,2,3,4,5,6,7,8,9]) == 1000020

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

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

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

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

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

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

Answer: assert Solution().minCost(nums = [1000000, 1, 2, 3]) == 1000002

Answer: assert Solution().minCost(nums = [1000000, 999999, 999998, 999997]) == 1999998

Answer: assert Solution().minCost(nums = [1000000]) == 1000000

Answer: assert Solution().minCost(nums = [2, 7, 1, 8, 2, 8, 1, 8, 2, 8, 4, 5, 9, 0]) == 38

Answer: assert Solution().minCost(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000]) == 3000

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

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

Answer: assert Solution().minCost(nums = [1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3]) == 23

Answer: assert Solution().minCost(nums = [100, 200, 50, 400, 300, 600, 700, 800, 900, 1000]) == 2800

Answer: assert Solution().minCost(nums = [1,9,2,8,3,7,4,6,5,10,11,12]) == 43

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

Answer: assert Solution().minCost(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]) == 2550

Answer: assert Solution().minCost(nums = [1000000, 999999, 999998, 999997, 999996, 999995, 999994]) == 3999988

Answer: assert Solution().minCost(nums = [999999, 1, 1000000, 2, 999998, 3, 999997, 4, 999996, 5]) == 3000000

Answer: assert Solution().minCost(nums = [15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]) == 64

Answer: assert Solution().minCost(nums = [1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5]) == 26

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

Answer: assert Solution().minCost(nums = [2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10]) == 84

Answer: assert Solution().minCost(nums = [100,100,100,99,99,99,98,98,98,97,97,97]) == 592

Answer: assert Solution().minCost(nums = [2, 3, 1, 5, 4, 6, 8, 7, 10, 9, 12, 11, 14, 13, 16, 15]) == 72

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

Answer: assert Solution().minCost(nums = [9, 1, 5, 7, 3, 8, 6]) == 25

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

Answer: assert Solution().minCost(nums = [5, 3, 8, 6, 2, 7, 4, 1]) == 22

Answer: assert Solution().minCost(nums = [100,90,80,70,60,50,40,30,20,10,9,8,7,6,5,4,3,2,1]) == 325

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

Answer: assert Solution().minCost(nums = [1000000,500000,250000,125000,62500,31250,15625]) == 1328125

Answer: assert Solution().minCost(nums = [5, 3, 8, 6, 7, 2, 4]) == 21

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

Answer: assert Solution().minCost(nums = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10]) == 50

Answer: assert Solution().minCost(nums = [10, 10, 10, 10, 10, 1, 1, 1, 1, 1]) == 32

Answer: assert Solution().minCost(nums = [1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5]) == 30

Answer: assert Solution().minCost(nums = [10,20,30,1,2,3,4,5,6,7,8,9]) == 60

Answer: assert Solution().minCost(nums = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) == 64

Answer: assert Solution().minCost(nums = [1, 10, 100, 1000, 10000, 100000, 1000000]) == 1010101

Answer: assert Solution().minCost(nums = [3,1,4,1,5,9,2,6,5,3,5,9]) == 30

Answer: assert Solution().minCost(nums = [1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1,2,3]) == 32

Answer: assert Solution().minCost(nums = [999, 1000, 998, 1001, 997, 1002, 996, 1003, 995, 1004]) == 5000

Answer: assert Solution().minCost(nums = [5, 10, 15, 20, 25, 30, 35, 40, 45, 50]) == 150

Answer: assert Solution().minCost(nums = [999999,999998,999997,999996,999995,999994,999993,999992,999991,999990]) == 4999975

Answer: assert Solution().minCost(nums = [10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10]) == 150

Answer: assert Solution().minCost(nums = [1,2,1,2,1,2,1,2,1,2]) == 8

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

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

Answer: assert Solution().minCost(nums = [5, 8, 2, 9, 3, 7, 6, 1, 4]) == 26

Answer: assert Solution().minCost(nums = [10,10,10,9,9,9,8,8,8,7,7,7,6,6,6,5,5,5,4,4,4,3,3,3,2,2,2,1,1,1]) == 85

Answer: assert Solution().minCost(nums = [5,3,8,1,9,2,7]) == 22

Answer: assert Solution().minCost(nums = [1000000, 999999, 999998, 999997, 999996, 999995, 999994, 999993, 999992, 999991]) == 4999980

Answer: assert Solution().minCost(nums = [5,5,5,5,5,5,5,5,5,5]) == 25

Answer: assert Solution().minCost(nums = [10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1]) == 59

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

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

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

Answer: assert Solution().minCost(nums = [5, 5, 5, 5, 5, 5, 5, 5]) == 20

Answer: assert Solution().minCost(nums = [100, 200, 150, 50, 300, 250, 1000, 750, 500, 250]) == 2100

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

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

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

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

Answer: assert Solution().minCost(nums = [10,20,30,40,50,60,70,80,90,100]) == 300

Answer: assert Solution().minCost(nums = [5, 1, 9, 3, 8, 2, 7, 4, 6, 10]) == 33

Answer: assert Solution().minCost(nums = [100,1,100,2,100,3,100,4,100,5,100,6,100,7,100,8,100,9,100,10]) == 530

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

Answer: assert Solution().minCost(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]) == 64

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

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

Answer: assert Solution().minCost(nums = [1, 3, 5, 7, 9, 11, 13, 15]) == 36

Answer: assert Solution().minCost(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200]) == 1100

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

Answer: assert Solution().minCost(nums = [3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5]) == 27

Answer: assert Solution().minCost(nums = [100, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) == 105

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

Answer: assert Solution().minCost(nums = [10,20,30,40,50,60,70,80,90,100,110,120,130,140,150]) == 640

Answer: assert Solution().minCost(nums = [500, 501, 499, 502, 498, 503, 497, 504, 496, 505]) == 2505

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

Answer: assert Solution().minCost(nums = [1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5]) == 24

Answer: assert Solution().minCost(nums = [1, 2, 3, 1000000, 4, 5, 6, 1000000, 7, 8, 9, 1000000, 10, 11, 12, 1000000]) == 2000042

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

Answer: assert Solution().minCost(nums = [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, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60]) == 1680

Answer: assert Solution().minCost(nums = [1,2,1000000,2,1,1000000,2,1,1000000,2,1,1000000,2,1,1000000]) == 3000007

Answer: assert Solution().minCost(nums = [10, 20, 30, 40, 50, 60, 70]) == 160

Answer: assert Solution().minCost(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]) == 93

Answer: assert Solution().minCost(nums = [1000, 1, 1000, 1, 1000, 1, 1000, 1]) == 2002

Answer: assert Solution().minCost(nums = [123456, 654321, 111111, 222222, 333333, 444444]) == 1111110

Answer: assert Solution().minCost(nums = [50, 40, 30, 20, 10, 5, 15, 25, 35, 45]) == 160

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

Answer: assert Solution().minCost(nums = [1, 10, 2, 9, 3, 8, 4, 7, 5, 6, 11, 12, 13, 14, 15]) == 68

Answer: assert Solution().minCost(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100]) == 300

Answer: assert Solution().minCost(nums = [1,1000000,2,999999,3,999998]) == 2000000

Answer: assert Solution().minCost(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) == 10

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

Answer: assert Solution().minCost(nums = [1000000,999999,999998,999997,999996,999995,999994,999993,999992,999991,999990,999989,999988,999987,999986,999985,999984,999983,999982,999981]) == 9999910

Answer: assert Solution().minCost(nums = [1000000, 1, 1000000, 1, 1000000, 1, 1000000, 1, 1000000]) == 3000002

Answer: assert Solution().minCost(nums = [1000000,1,1000000,1,1000000,1]) == 2000001

Answer: assert Solution().minCost(nums = [500000, 400000, 300000, 200000, 100000, 90000, 80000, 70000, 60000, 50000, 40000, 30000, 20000, 10000, 9000, 8000, 7000, 6000, 5000, 4000, 3000, 2000, 1000, 900, 800, 700, 600, 500, 400, 300, 200, 100, 90, 80, 70, 60, 50, 40, 30, 20, 10, 1]) == 1127250

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, nums: list[int]) -> int:
    n = len(nums)

    @functools.lru_cache(None)
    def dp(last: int, i: int) -> int:
      if i == n:  # Single element left.
        return nums[last]
      if i == n - 1:  # Two elements left.
        return max(nums[last], nums[i])
      a = max(nums[i], nums[i + 1]) + dp(last, i + 2)
      b = max(nums[last], nums[i]) + dp(i + 1, i + 2)
      c = max(nums[last], nums[i + 1]) + dp(i, i + 2)
      return min(a, b, c)

    return dp(0, 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:
Given two integers, n and k, an alternating permutation is a permutation of the first n positive integers such that no two adjacent elements are both odd or both even.
Return the k-th alternating permutation sorted in lexicographical order. If there are fewer than k valid alternating permutations, return an empty list.

Example 1:

Input: n = 4, k = 6
Output: [3,4,1,2]
Explanation:
The lexicographically-sorted alternating permutations of [1, 2, 3, 4] are:

[1, 2, 3, 4]
[1, 4, 3, 2]
[2, 1, 4, 3]
[2, 3, 4, 1]
[3, 2, 1, 4]
[3, 4, 1, 2] ← 6th permutation
[4, 1, 2, 3]
[4, 3, 2, 1]

Since k = 6, we return [3, 4, 1, 2].

Example 2:

Input: n = 3, k = 2
Output: [3,2,1]
Explanation:
The lexicographically-sorted alternating permutations of [1, 2, 3] are:

[1, 2, 3]
[3, 2, 1] ← 2nd permutation

Since k = 2, we return [3, 2, 1].

Example 3:

Input: n = 2, k = 3
Output: []
Explanation:
The lexicographically-sorted alternating permutations of [1, 2] are:

[1, 2]
[2, 1]

There are only 2 alternating permutations, but k = 3, which is out of range. Thus, we return an empty list [].

Constraints:

1 <= n <= 100
1 <= k <= 1015

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

SOLUTION:

Question:
PROBLEM:
Given two integers, n and k, an alternating permutation is a permutation of the first n positive integers such that no two adjacent elements are both odd or both even.
Return the k-th alternating permutation sorted in lexicographical order. If there are fewer than k valid alternating permutations, return an empty list.

Example 1:

Input: n = 4, k = 6
Output: [3,4,1,2]
Explanation:
The lexicographically-sorted alternating permutations of [1, 2, 3, 4] are:

[1, 2, 3, 4]
[1, 4, 3, 2]
[2, 1, 4, 3]
[2, 3, 4, 1]
[3, 2, 1, 4]
[3, 4, 1, 2] ← 6th permutation
[4, 1, 2, 3]
[4, 3, 2, 1]

Since k = 6, we return [3, 4, 1, 2].

Example 2:

Input: n = 3, k = 2
Output: [3,2,1]
Explanation:
The lexicographically-sorted alternating permutations of [1, 2, 3] are:

[1, 2, 3]
[3, 2, 1] ← 2nd permutation

Since k = 2, we return [3, 2, 1].

Example 3:

Input: n = 2, k = 3
Output: []
Explanation:
The lexicographically-sorted alternating permutations of [1, 2] are:

[1, 2]
[2, 1]

There are only 2 alternating permutations, but k = 3, which is out of range. Thus, we return an empty list [].

Constraints:

1 <= n <= 100
1 <= k <= 1015

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

SOLUTION:

assert Solution().permute(n = 10, k = 1000000) == []

assert Solution().permute(n = 3, k = 2) == [3, 2, 1]

assert Solution().permute(n = 1, k = 1) == [1]

assert Solution().permute(n = 5, k = 1) == [1, 2, 3, 4, 5]

assert Solution().permute(n = 9, k = 5040) == []

assert Solution().permute(n = 5, k = 20) == []

assert Solution().permute(n = 6, k = 100) == []

assert Solution().permute(n = 10, k = 3628800) == []

assert Solution().permute(n = 2, k = 3) == []

assert Solution().permute(n = 7, k = 5000) == []

assert Solution().permute(n = 6, k = 120) == []

assert Solution().permute(n = 4, k = 6) == [3, 4, 1, 2]

assert Solution().permute(n = 8, k = 10000) == []

assert Solution().permute(n = 9, k = 500000) == []

assert Solution().permute(n = 9, k = 123456) == []

assert Solution().permute(n = 11, k = 1000000) == []

assert Solution().permute(n = 10, k = 987654321) == []

assert Solution().permute(n = 12, k = 10000000) == []

assert Solution().permute(n = 13, k = 50000000) == []

assert Solution().permute(n = 8, k = 40320) == []

assert Solution().permute(n = 9, k = 35000) == []

assert Solution().permute(n = 10, k = 500000) == []

assert Solution().permute(n = 9, k = 20480) == []

assert Solution().permute(n = 9, k = 345678) == []

assert Solution().permute(n = 11, k = 9876543) == []

assert Solution().permute(n = 9, k = 1000000) == []

assert Solution().permute(n = 10, k = 2000000) == []

assert Solution().permute(n = 9, k = 362880) == []

assert Solution().permute(n = 10, k = 500000000) == []

assert Solution().permute(n = 5, k = 120) == []

assert Solution().permute(n = 10, k = 3456789) == []

assert Solution().permute(n = 10, k = 50000) == []

assert Solution().permute(n = 12, k = 100000000) == []

assert Solution().permute(n = 13, k = 111111111) == []

assert Solution().permute(n = 14, k = 999999999) == []

assert Solution().permute(n = 9, k = 100000) == []

assert Solution().permute(n = 9, k = 50000) == []

assert Solution().permute(n = 8, k = 1000000) == []

Answer: assert Solution().permute(n = 100, k = 1000000000000000) == [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, 91, 98, 99, 88, 79, 90, 93, 96, 97, 82, 89, 80, 85, 86, 81, 92, 95, 84, 87, 100, 83, 94]

Answer: assert Solution().permute(n = 10, k = 1000000) == []

Answer: assert Solution().permute(n = 3, k = 2) == [3, 2, 1]

Answer: assert Solution().permute(n = 1, k = 1) == [1]

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

Answer: assert Solution().permute(n = 10, k = 100) == [1, 2, 3, 8, 9, 4, 7, 10, 5, 6]

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

Answer: assert Solution().permute(n = 9, k = 5040) == []

Answer: assert Solution().permute(n = 5, k = 20) == []

Answer: assert Solution().permute(n = 6, k = 100) == []

Answer: assert Solution().permute(n = 8, k = 1000) == [7, 8, 5, 2, 3, 6, 1, 4]

Answer: assert Solution().permute(n = 7, k = 100) == [5, 6, 1, 4, 7, 2, 3]

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

Answer: assert Solution().permute(n = 10, k = 500) == [1, 2, 9, 6, 7, 8, 5, 10, 3, 4]

Answer: assert Solution().permute(n = 10, k = 3628800) == []

Answer: assert Solution().permute(n = 2, k = 3) == []

Answer: assert Solution().permute(n = 7, k = 5000) == []

Answer: assert Solution().permute(n = 6, k = 120) == []

Answer: assert Solution().permute(n = 6, k = 20) == [2, 3, 6, 5, 4, 1]

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

Answer: assert Solution().permute(n = 100, k = 1) == [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]

Answer: assert Solution().permute(n = 8, k = 10000) == []

Answer: assert Solution().permute(n = 10, k = 10000) == [4, 5, 6, 3, 10, 1, 8, 9, 2, 7]

Answer: assert Solution().permute(n = 9, k = 500000) == []

Answer: assert Solution().permute(n = 9, k = 123456) == []

Answer: assert Solution().permute(n = 50, k = 10000000000) == [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, 46, 37, 36, 45, 48, 35, 44, 43, 38, 39, 40, 49, 34, 47, 50, 41, 42]

Answer: assert Solution().permute(n = 11, k = 1000000) == []

Answer: assert Solution().permute(n = 75, k = 5000000) == [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, 64, 67, 70, 75, 66, 63, 72, 65, 74, 71, 68, 73, 62, 69]

Answer: assert Solution().permute(n = 12, k = 50000) == [1, 8, 7, 4, 11, 2, 9, 10, 5, 12, 3, 6]

Answer: assert Solution().permute(n = 99, k = 1) == [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]

Answer: assert Solution().permute(n = 10, k = 987654321) == []

Answer: assert Solution().permute(n = 20, k = 100000000) == [1, 2, 3, 4, 5, 12, 19, 14, 17, 10, 11, 8, 7, 16, 15, 6, 13, 20, 9, 18]

Answer: assert Solution().permute(n = 12, k = 1000000) == [12, 7, 6, 3, 2, 5, 10, 1, 8, 11, 4, 9]

Answer: assert Solution().permute(n = 18, k = 750000) == [1, 2, 3, 4, 5, 8, 11, 16, 7, 12, 9, 10, 13, 18, 17, 14, 15, 6]

Answer: assert Solution().permute(n = 12, k = 10000000) == []

Answer: assert Solution().permute(n = 13, k = 50000000) == []

Answer: assert Solution().permute(n = 50, k = 123456789) == [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, 44, 49, 36, 37, 50, 41, 48, 45, 46, 39, 42, 43, 38, 47, 40]

Answer: assert Solution().permute(n = 15, k = 100000000) == [7, 14, 9, 12, 5, 6, 3, 2, 11, 10, 1, 8, 15, 4, 13]

Answer: assert Solution().permute(n = 50, k = 10000000) == [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, 41, 48, 39, 46, 45, 40, 37, 42, 49, 38, 47, 50, 43, 44]

Answer: assert Solution().permute(n = 8, k = 40320) == []

Answer: assert Solution().permute(n = 9, k = 35000) == []

Answer: assert Solution().permute(n = 8, k = 1024) == [8, 1, 4, 3, 6, 7, 2, 5]

Answer: assert Solution().permute(n = 16, k = 10000000000) == []

Answer: assert Solution().permute(n = 10, k = 500000) == []

Answer: assert Solution().permute(n = 15, k = 10000000) == [1, 6, 13, 4, 11, 10, 5, 2, 7, 14, 3, 12, 15, 8, 9]

Answer: assert Solution().permute(n = 9, k = 20480) == []

Answer: assert Solution().permute(n = 20, k = 2432902008176640000) == []

Answer: assert Solution().permute(n = 9, k = 345678) == []

Answer: assert Solution().permute(n = 11, k = 9876543) == []

Answer: assert Solution().permute(n = 12, k = 500000) == [6, 9, 10, 7, 2, 1, 12, 5, 8, 11, 4, 3]

Answer: assert Solution().permute(n = 100, k = 100000000000000) == [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, 94, 91, 100, 87, 90, 97, 98, 99, 82, 89, 86, 85, 84, 81, 88, 95, 80, 93, 96, 83, 92]

Answer: assert Solution().permute(n = 9, k = 1000000) == []

Answer: assert Solution().permute(n = 100, k = 999999999999) == [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, 96, 93, 88, 83, 100, 95, 84, 89, 82, 91, 90, 85, 92, 99, 86, 97, 94, 87, 98]

Answer: assert Solution().permute(n = 10, k = 2000000) == []

Answer: assert Solution().permute(n = 9, k = 362880) == []

Answer: assert Solution().permute(n = 25, k = 500000) == [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 24, 23, 20, 21, 14, 15, 22, 19, 18, 25, 16, 17]

Answer: assert Solution().permute(n = 10, k = 500000000) == []

Answer: assert Solution().permute(n = 99, k = 50000000000000) == [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, 85, 94, 99, 92, 83, 98, 89, 96, 93, 82, 95, 84, 79, 88, 81, 90, 91, 86, 97, 80, 87]

Answer: assert Solution().permute(n = 12, k = 10000) == [1, 2, 9, 8, 5, 6, 11, 4, 7, 12, 3, 10]

Answer: assert Solution().permute(n = 15, k = 123456789) == [9, 14, 1, 2, 15, 6, 13, 10, 11, 4, 7, 8, 3, 12, 5]

Answer: assert Solution().permute(n = 5, k = 120) == []

Answer: assert Solution().permute(n = 90, k = 1000000) == [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, 80, 89, 86, 83, 82, 79, 84, 87, 78, 85, 90, 81, 88]

Answer: assert Solution().permute(n = 30, k = 500000000000) == [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 18, 27, 14, 21, 22, 25, 26, 15, 12, 17, 16, 23, 20, 29, 28, 19, 30, 13, 24]

Answer: assert Solution().permute(n = 10, k = 1024) == [1, 4, 9, 2, 5, 6, 7, 10, 3, 8]

Answer: assert Solution().permute(n = 15, k = 5000000) == [1, 4, 7, 10, 15, 6, 3, 12, 5, 14, 11, 8, 13, 2, 9]

Answer: assert Solution().permute(n = 15, k = 50000) == [1, 2, 3, 4, 11, 10, 7, 14, 5, 12, 13, 8, 15, 6, 9]

Answer: assert Solution().permute(n = 10, k = 3456789) == []

Answer: assert Solution().permute(n = 25, k = 1000) == [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 22, 25, 24, 17, 20, 23, 18, 21]

Answer: assert Solution().permute(n = 10, k = 50000) == []

Answer: assert Solution().permute(n = 100, k = 10000000000000) == [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, 95, 92, 99, 90, 83, 88, 97, 100, 87, 96, 89, 84, 81, 86, 93, 82, 91, 98, 85, 94]

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

Answer: assert Solution().permute(n = 50, k = 1000000000) == [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, 43, 50, 39, 44, 35, 36, 45, 40, 37, 42, 49, 38, 47, 48, 41, 46]

Answer: assert Solution().permute(n = 25, k = 10000000000) == [1, 2, 3, 4, 5, 6, 7, 8, 21, 12, 11, 20, 23, 10, 19, 18, 13, 14, 15, 24, 9, 22, 25, 16, 17]

Answer: assert Solution().permute(n = 12, k = 100000000) == []

Answer: assert Solution().permute(n = 50, k = 1000000000000) == [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, 46, 43, 38, 33, 50, 45, 34, 39, 32, 41, 40, 35, 42, 49, 36, 47, 48, 37, 44]

Answer: assert Solution().permute(n = 13, k = 111111111) == []

Answer: assert Solution().permute(n = 15, k = 12345) == [1, 2, 3, 4, 5, 14, 9, 8, 13, 12, 15, 6, 7, 10, 11]

Answer: assert Solution().permute(n = 18, k = 500000000) == [1, 2, 7, 10, 13, 16, 11, 4, 5, 6, 15, 8, 17, 14, 9, 18, 3, 12]

Answer: assert Solution().permute(n = 18, k = 500000) == [1, 2, 3, 4, 5, 6, 17, 16, 13, 14, 7, 8, 15, 12, 11, 18, 9, 10]

Answer: assert Solution().permute(n = 100, k = 1000000000000) == [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, 96, 93, 88, 83, 100, 95, 84, 89, 82, 91, 90, 85, 92, 99, 86, 97, 98, 87, 94]

Answer: assert Solution().permute(n = 14, k = 999999999) == []

Answer: assert Solution().permute(n = 20, k = 999999999999999) == []

Answer: assert Solution().permute(n = 50, k = 100) == [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, 48, 49, 44, 47, 50, 45, 46]

Answer: assert Solution().permute(n = 15, k = 1000000) == [1, 2, 5, 14, 11, 8, 7, 4, 9, 12, 3, 10, 15, 6, 13]

Answer: assert Solution().permute(n = 20, k = 100000) == [1, 2, 3, 4, 5, 6, 7, 8, 11, 10, 19, 18, 15, 14, 17, 12, 13, 20, 9, 16]

Answer: assert Solution().permute(n = 75, k = 5) == [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, 73, 72, 71, 74, 75]

Answer: assert Solution().permute(n = 100, k = 500000000000000) == [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, 85, 88, 99, 96, 79, 84, 87, 100, 91, 80, 97, 82, 83, 86, 93, 90, 95, 94, 89, 98, 81, 92]

Answer: assert Solution().permute(n = 100, k = 2000000000000000000) == [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, 92, 93, 90, 81, 84, 79, 94, 97, 96, 85, 98, 95, 88, 77, 76, 99, 82, 83, 100, 89, 80, 91, 86, 87, 78]

Answer: assert Solution().permute(n = 15, k = 1000000000) == []

Answer: assert Solution().permute(n = 14, k = 1000000000) == []

Answer: assert Solution().permute(n = 9, k = 100000) == []

Answer: assert Solution().permute(n = 10, k = 1000) == [1, 4, 7, 10, 9, 2, 5, 8, 3, 6]

Answer: assert Solution().permute(n = 9, k = 50000) == []

Answer: assert Solution().permute(n = 20, k = 1000000) == [1, 2, 3, 4, 5, 6, 7, 10, 19, 16, 13, 12, 9, 14, 17, 8, 15, 20, 11, 18]

Answer: assert Solution().permute(n = 8, k = 1000000) == []

Answer: assert Solution().permute(n = 10, k = 5000) == [2, 7, 8, 5, 10, 3, 6, 9, 4, 1]

Answer: assert Solution().permute(n = 15, k = 10000) == [1, 2, 3, 4, 5, 12, 11, 8, 9, 14, 7, 10, 15, 6, 13]

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 permute(self, n: int, k: int) -> list[int]:
    ans = []
    isLookingForEven = True
    remainingNumbers = list(range(1, n + 1))

    for turn in range(n):
      remainingPermutations = (math.factorial((n - 1 - turn) // 2) *
                               math.factorial((n - turn) // 2))
      found = False
      for index, number in enumerate(remainingNumbers):
        if number % 2 != isLookingForEven and (turn > 0 or n % 2 == 1):
          continue
        if k <= remainingPermutations:
          ans.append(remainingNumbers.pop(index))
          isLookingForEven = ans[-1] % 2 == 0
          found = True
          break
        k -= remainingPermutations
      if not found:
        return []

    return ans


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

PROBLEM:
You are given a string s and an integer k.
In one operation, you can replace the character at any position with the next or previous letter in the alphabet (wrapping around so that 'a' is after 'z'). For example, replacing 'a' with the next letter results in 'b', and replacing 'a' with the previous letter results in 'z'. Similarly, replacing 'z' with the next letter results in 'a', and replacing 'z' with the previous letter results in 'y'.
Return the length of the longest palindromic subsequence of s that can be obtained after performing at most k operations.

Example 1:

Input: s = "abced", k = 2
Output: 3
Explanation:

Replace s[1] with the next letter, and s becomes "acced".
Replace s[4] with the previous letter, and s becomes "accec".

The subsequence "ccc" forms a palindrome of length 3, which is the maximum.

Example 2:

Input: s = "aaazzz", k = 4
Output: 6
Explanation:

Replace s[0] with the previous letter, and s becomes "zaazzz".
Replace s[4] with the next letter, and s becomes "zaazaz".
Replace s[3] with the next letter, and s becomes "zaaaaz".

The entire string forms a palindrome of length 6.

Constraints:

1 <= s.length <= 200
1 <= k <= 200
s consists of only lowercase English letters.

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

SOLUTION:

Question:
PROBLEM:
You are given a string s and an integer k.
In one operation, you can replace the character at any position with the next or previous letter in the alphabet (wrapping around so that 'a' is after 'z'). For example, replacing 'a' with the next letter results in 'b', and replacing 'a' with the previous letter results in 'z'. Similarly, replacing 'z' with the next letter results in 'a', and replacing 'z' with the previous letter results in 'y'.
Return the length of the longest palindromic subsequence of s that can be obtained after performing at most k operations.

Example 1:

Input: s = "abced", k = 2
Output: 3
Explanation:

Replace s[1] with the next letter, and s becomes "acced".
Replace s[4] with the previous letter, and s becomes "accec".

The subsequence "ccc" forms a palindrome of length 3, which is the maximum.

Example 2:

Input: s = "aaazzz", k = 4
Output: 6
Explanation:

Replace s[0] with the previous letter, and s becomes "zaazzz".
Replace s[4] with the next letter, and s becomes "zaazaz".
Replace s[3] with the next letter, and s becomes "zaaaaz".

The entire string forms a palindrome of length 6.

Constraints:

1 <= s.length <= 200
1 <= k <= 200
s consists of only lowercase English letters.

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

SOLUTION:

assert Solution().longestPalindromicSubsequence(s = "zxy", k = 1) == 3

assert Solution().longestPalindromicSubsequence(s = "a", k = 5) == 1

assert Solution().longestPalindromicSubsequence(s = "abced", k = 2) == 3

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyz", k = 25) == 14

assert Solution().longestPalindromicSubsequence(s = "zzzzz", k = 1) == 5

assert Solution().longestPalindromicSubsequence(s = "abcdeedcba", k = 5) == 10

assert Solution().longestPalindromicSubsequence(s = "zzzzz", k = 10) == 5

assert Solution().longestPalindromicSubsequence(s = "abcba", k = 0) == 5

assert Solution().longestPalindromicSubsequence(s = "aaazzz", k = 4) == 6

assert Solution().longestPalindromicSubsequence(s = "abcdef", k = 3) == 3

assert Solution().longestPalindromicSubsequence(s = "adadd", k = 2) == 3

assert Solution().longestPalindromicSubsequence(s = "mamad", k = 3) == 3

assert Solution().longestPalindromicSubsequence(s = "zyxwvutsrqponmlkjihgfedcba", k = 200) == 26

assert Solution().longestPalindromicSubsequence(s = "xyz", k = 1) == 2

assert Solution().longestPalindromicSubsequence(s = "abba", k = 1) == 4

assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 0) == 9

assert Solution().longestPalindromicSubsequence(s = "aabaa", k = 2) == 5

assert Solution().longestPalindromicSubsequence(s = "mnopqr", k = 10) == 6

assert Solution().longestPalindromicSubsequence(s = "a", k = 0) == 1

assert Solution().longestPalindromicSubsequence(s = "abcdefg", k = 3) == 3

assert Solution().longestPalindromicSubsequence(s = "abcdabcdabcd", k = 6) == 11

assert Solution().longestPalindromicSubsequence(s = "a", k = 1) == 1

assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 5) == 9

assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzz", k = 0) == 10

assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzz", k = 100) == 20

assert Solution().longestPalindromicSubsequence(s = "racecar", k = 10) == 7

assert Solution().longestPalindromicSubsequence(s = "zzyzxzyzxzyzxzyzxzyz", k = 30) == 20

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzz", k = 50) == 20

assert Solution().longestPalindromicSubsequence(s = "aaaabbbb", k = 5) == 8

assert Solution().longestPalindromicSubsequence(s = "abcabcabcabcabcabcabcabcabcabc", k = 25) == 30

assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbbcccc", k = 5) == 10

assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbbaaaa", k = 50) == 14

assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbbaaaaa", k = 10) == 15

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 180) == 51

assert Solution().longestPalindromicSubsequence(s = "level", k = 5) == 5

assert Solution().longestPalindromicSubsequence(s = "racecar", k = 5) == 7

assert Solution().longestPalindromicSubsequence(s = "abcdefgabcdefgabcdefgabcdefg", k = 10) == 17

assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 1) == 9

assert Solution().longestPalindromicSubsequence(s = "qwertyuioplkjhgfdsazxcvbnmqwertyuioplkjhgfdsazxcvb", k = 100) == 47

assert Solution().longestPalindromicSubsequence(s = "abxyzabcxyzabcxyz", k = 25) == 17

assert Solution().longestPalindromicSubsequence(s = "level", k = 1) == 5

assert Solution().longestPalindromicSubsequence(s = "abcba", k = 1) == 5

assert Solution().longestPalindromicSubsequence(s = "ababababababababababababababababababab", k = 75) == 38

assert Solution().longestPalindromicSubsequence(s = "abcdefgxyzzyxgfedcba", k = 15) == 20

assert Solution().longestPalindromicSubsequence(s = "refer", k = 0) == 5

assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffggzzzyyxxwwvvuuttsrqponmlkjihgfedcba", k = 100) == 47

assert Solution().longestPalindromicSubsequence(s = "palindrome", k = 15) == 9

assert Solution().longestPalindromicSubsequence(s = "amanaplanacanalpanama", k = 100) == 21

assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnm", k = 50) == 23

assert Solution().longestPalindromicSubsequence(s = "aaaabbbbcccc", k = 15) == 12

assert Solution().longestPalindromicSubsequence(s = "mnonmonmomnonmonmonmomnonm", k = 15) == 26

assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 7) == 10

assert Solution().longestPalindromicSubsequence(s = "banana", k = 10) == 5

assert Solution().longestPalindromicSubsequence(s = "aabacbebebe", k = 5) == 8

assert Solution().longestPalindromicSubsequence(s = "madamimadam", k = 10) == 11

assert Solution().longestPalindromicSubsequence(s = "abcdexyzabcdexyzabcdexyz", k = 50) == 24

assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 10) == 10

assert Solution().longestPalindromicSubsequence(s = "deified", k = 4) == 7

assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 2) == 9

assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 50) == 28

assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 3) == 9

assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzyxwvutsrqponmlkjihgfedcbazyxwvutsrqponmlkjihgfedcba", k = 200) == 60

assert Solution().longestPalindromicSubsequence(s = "aabbccabcabcabcabcabc", k = 50) == 21

assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 1) == 50

assert Solution().longestPalindromicSubsequence(s = "abcdeedcba", k = 0) == 10

assert Solution().longestPalindromicSubsequence(s = "abbaacddcabb", k = 4) == 11

assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 200) == 52

assert Solution().longestPalindromicSubsequence(s = "zzzyzzyzzyzzyzzyzzyz", k = 200) == 20

assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 5) == 9

assert Solution().longestPalindromicSubsequence(s = "level", k = 2) == 5

assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 0) == 32

assert Solution().longestPalindromicSubsequence(s = "ababbabbbabaaa", k = 15) == 14

assert Solution().longestPalindromicSubsequence(s = "rotor", k = 15) == 5

assert Solution().longestPalindromicSubsequence(s = "noon", k = 1) == 4

assert Solution().longestPalindromicSubsequence(s = "noonhighnoon", k = 5) == 12

assert Solution().longestPalindromicSubsequence(s = "abcdbca", k = 2) == 7

assert Solution().longestPalindromicSubsequence(s = "abcdxyzabcdxyzabcdxyz", k = 30) == 21

assert Solution().longestPalindromicSubsequence(s = "banana", k = 3) == 5

assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnmnbvcxzlkjhgfdsapoiuytrewq", k = 50) == 51

assert Solution().longestPalindromicSubsequence(s = "abacbacbacbacbacbacbacbacbacbacbacbacbacbacbacbacb", k = 20) == 50

assert Solution().longestPalindromicSubsequence(s = "abacaba", k = 3) == 7

assert Solution().longestPalindromicSubsequence(s = "mamamamamamamamamamamamamamamamama", k = 50) == 33

assert Solution().longestPalindromicSubsequence(s = "racecar", k = 2) == 7

assert Solution().longestPalindromicSubsequence(s = "noon", k = 2) == 4

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 250) == 51

assert Solution().longestPalindromicSubsequence(s = "abcdefg", k = 6) == 5

assert Solution().longestPalindromicSubsequence(s = "zyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 300) == 84

assert Solution().longestPalindromicSubsequence(s = "madam", k = 1) == 5

assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyx", k = 100) == 57

assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyx", k = 200) == 39

assert Solution().longestPalindromicSubsequence(s = "rotor", k = 0) == 5

assert Solution().longestPalindromicSubsequence(s = "abcdefgzyxwvutsrqponmlkjihgfedcba", k = 50) == 29

assert Solution().longestPalindromicSubsequence(s = "abcdefghzyxwvutsrqponmlkjihgfedcba", k = 150) == 34

assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnmqwertyuiopasdfghjklzxcvbnm", k = 100) == 47

assert Solution().longestPalindromicSubsequence(s = "abcdefghizyxwvutsrqponmlkjihgfedcba", k = 100) == 35

assert Solution().longestPalindromicSubsequence(s = "deified", k = 2) == 7

assert Solution().longestPalindromicSubsequence(s = "aaaabbbbccccdddd", k = 10) == 13

assert Solution().longestPalindromicSubsequence(s = "madam", k = 2) == 5

assert Solution().longestPalindromicSubsequence(s = "rotor", k = 3) == 5

assert Solution().longestPalindromicSubsequence(s = "abcdefgihgfedcb", k = 15) == 15

assert Solution().longestPalindromicSubsequence(s = "abcdefghihgfedcba", k = 10) == 17

assert Solution().longestPalindromicSubsequence(s = "pqrstuvwxyzabcdefghijklmno", k = 50) == 20

assert Solution().longestPalindromicSubsequence(s = "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", k = 0) == 30

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyz", k = 10) == 9

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxzy", k = 100) == 26

assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzz", k = 10) == 20

assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 150) == 49

assert Solution().longestPalindromicSubsequence(s = "abcabcabcabcabcabcabcabcabcabc", k = 100) == 30

assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 5) == 52

assert Solution().longestPalindromicSubsequence(s = "reviled", k = 7) == 5

assert Solution().longestPalindromicSubsequence(s = "abccba", k = 0) == 6

assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyx", k = 100) == 21

assert Solution().longestPalindromicSubsequence(s = "pneumonoultramicroscopicsilicovolcanoconiosis", k = 50) == 37

assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyz", k = 10) == 6

assert Solution().longestPalindromicSubsequence(s = "step on no pets", k = 10) == 15

assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyx", k = 100) == 42

assert Solution().longestPalindromicSubsequence(s = "abababababababababababababababab", k = 50) == 32

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyz", k = 50) == 20

assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 3) == 9

assert Solution().longestPalindromicSubsequence(s = "abacabadabacaba", k = 5) == 15

assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzzyxwvutsrqponmlk", k = 80) == 30

assert Solution().longestPalindromicSubsequence(s = "abcdezyxwvutsrqponmlkjihgf", k = 50) == 20

assert Solution().longestPalindromicSubsequence(s = "abacabadabacaba", k = 10) == 15

assert Solution().longestPalindromicSubsequence(s = "abcabcabc", k = 4) == 8

assert Solution().longestPalindromicSubsequence(s = "abababababababababab", k = 10) == 20

assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 100) == 40

assert Solution().longestPalindromicSubsequence(s = "abcabcabcabcabcabcabc", k = 20) == 21

assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnm", k = 25) == 18

assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbb", k = 10) == 10

assert Solution().longestPalindromicSubsequence(s = "zabacabadabacaba", k = 10) == 15

assert Solution().longestPalindromicSubsequence(s = "abcdefedcbaabcdefedcbaabcdefedcba", k = 60) == 33

assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 150) == 36

assert Solution().longestPalindromicSubsequence(s = "abcdxyzzyxcba", k = 10) == 13

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 100) == 51

assert Solution().longestPalindromicSubsequence(s = "kayak", k = 20) == 5

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyz", k = 100) == 40

assert Solution().longestPalindromicSubsequence(s = "abcdzdcba", k = 3) == 9

assert Solution().longestPalindromicSubsequence(s = "aababaaa", k = 2) == 8

assert Solution().longestPalindromicSubsequence(s = "deed", k = 3) == 4

assert Solution().longestPalindromicSubsequence(s = "ababababababababababababababababababababababababababab", k = 100) == 54

assert Solution().longestPalindromicSubsequence(s = "abcdefghijkmlkjihgfedcba", k = 50) == 24

assert Solution().longestPalindromicSubsequence(s = "abcdeedcba", k = 3) == 10

assert Solution().longestPalindromicSubsequence(s = "abcdefg", k = 10) == 6

assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 50) == 30

assert Solution().longestPalindromicSubsequence(s = "racecar", k = 0) == 7

assert Solution().longestPalindromicSubsequence(s = "zyxwvutsrqponmlkjihgfedcba", k = 150) == 26

assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 50) == 51

assert Solution().longestPalindromicSubsequence(s = "abcdefghijabcdefghij", k = 15) == 13

assert Solution().longestPalindromicSubsequence(s = "abacabadabacabadabacaba", k = 20) == 23

assert Solution().longestPalindromicSubsequence(s = "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", k = 5) == 52

assert Solution().longestPalindromicSubsequence(s = "abcdabcdabcdabcd", k = 20) == 16

assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 180) == 52

assert Solution().longestPalindromicSubsequence(s = "abcdefghijjihgfedcba", k = 5) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "zxy", k = 1) == 3

Answer: assert Solution().longestPalindromicSubsequence(s = "a", k = 5) == 1

Answer: assert Solution().longestPalindromicSubsequence(s = "abced", k = 2) == 3

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyz", k = 25) == 14

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzz", k = 1) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdeedcba", k = 5) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzz", k = 10) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "abcba", k = 0) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "aaazzz", k = 4) == 6

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdef", k = 3) == 3

Answer: assert Solution().longestPalindromicSubsequence(s = "adadd", k = 2) == 3

Answer: assert Solution().longestPalindromicSubsequence(s = "mamad", k = 3) == 3

Answer: assert Solution().longestPalindromicSubsequence(s = "zyxwvutsrqponmlkjihgfedcba", k = 200) == 26

Answer: assert Solution().longestPalindromicSubsequence(s = "xyz", k = 1) == 2

Answer: assert Solution().longestPalindromicSubsequence(s = "abba", k = 1) == 4

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 0) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "aabaa", k = 2) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "mnopqr", k = 10) == 6

Answer: assert Solution().longestPalindromicSubsequence(s = "a", k = 0) == 1

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefg", k = 3) == 3

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdabcdabcd", k = 6) == 11

Answer: assert Solution().longestPalindromicSubsequence(s = "a", k = 1) == 1

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 5) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzz", k = 0) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzz", k = 100) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "racecar", k = 10) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "zzyzxzyzxzyzxzyzxzyz", k = 30) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzz", k = 50) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaabbbb", k = 5) == 8

Answer: assert Solution().longestPalindromicSubsequence(s = "abcabcabcabcabcabcabcabcabcabc", k = 25) == 30

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbbcccc", k = 5) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbbaaaa", k = 50) == 14

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbbaaaaa", k = 10) == 15

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 180) == 51

Answer: assert Solution().longestPalindromicSubsequence(s = "level", k = 5) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "racecar", k = 5) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefgabcdefgabcdefgabcdefg", k = 10) == 17

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 1) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "qwertyuioplkjhgfdsazxcvbnmqwertyuioplkjhgfdsazxcvb", k = 100) == 47

Answer: assert Solution().longestPalindromicSubsequence(s = "abxyzabcxyzabcxyz", k = 25) == 17

Answer: assert Solution().longestPalindromicSubsequence(s = "level", k = 1) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "abcba", k = 1) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "ababababababababababababababababababab", k = 75) == 38

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefgxyzzyxgfedcba", k = 15) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "refer", k = 0) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffggzzzyyxxwwvvuuttsrqponmlkjihgfedcba", k = 100) == 47

Answer: assert Solution().longestPalindromicSubsequence(s = "palindrome", k = 15) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "amanaplanacanalpanama", k = 100) == 21

Answer: assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnm", k = 50) == 23

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaabbbbcccc", k = 15) == 12

Answer: assert Solution().longestPalindromicSubsequence(s = "mnonmonmomnonmonmonmomnonm", k = 15) == 26

Answer: assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 7) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "banana", k = 10) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "aabacbebebe", k = 5) == 8

Answer: assert Solution().longestPalindromicSubsequence(s = "madamimadam", k = 10) == 11

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdexyzabcdexyzabcdexyz", k = 50) == 24

Answer: assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 10) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "deified", k = 4) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 2) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 50) == 28

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdedcba", k = 3) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzyxwvutsrqponmlkjihgfedcbazyxwvutsrqponmlkjihgfedcba", k = 200) == 60

Answer: assert Solution().longestPalindromicSubsequence(s = "aabbccabcabcabcabcabc", k = 50) == 21

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 1) == 50

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdeedcba", k = 0) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "abbaacddcabb", k = 4) == 11

Answer: assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 200) == 52

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzyzzyzzyzzyzzyzzyz", k = 200) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 5) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "level", k = 2) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 0) == 32

Answer: assert Solution().longestPalindromicSubsequence(s = "ababbabbbabaaa", k = 15) == 14

Answer: assert Solution().longestPalindromicSubsequence(s = "rotor", k = 15) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "noon", k = 1) == 4

Answer: assert Solution().longestPalindromicSubsequence(s = "noonhighnoon", k = 5) == 12

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdbca", k = 2) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdxyzabcdxyzabcdxyz", k = 30) == 21

Answer: assert Solution().longestPalindromicSubsequence(s = "banana", k = 3) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnmnbvcxzlkjhgfdsapoiuytrewq", k = 50) == 51

Answer: assert Solution().longestPalindromicSubsequence(s = "abacbacbacbacbacbacbacbacbacbacbacbacbacbacbacbacb", k = 20) == 50

Answer: assert Solution().longestPalindromicSubsequence(s = "abacaba", k = 3) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "mamamamamamamamamamamamamamamamama", k = 50) == 33

Answer: assert Solution().longestPalindromicSubsequence(s = "racecar", k = 2) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "noon", k = 2) == 4

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 250) == 51

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefg", k = 6) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "zyzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 300) == 84

Answer: assert Solution().longestPalindromicSubsequence(s = "madam", k = 1) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyx", k = 100) == 57

Answer: assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyx", k = 200) == 39

Answer: assert Solution().longestPalindromicSubsequence(s = "rotor", k = 0) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefgzyxwvutsrqponmlkjihgfedcba", k = 50) == 29

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghzyxwvutsrqponmlkjihgfedcba", k = 150) == 34

Answer: assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnmqwertyuiopasdfghjklzxcvbnm", k = 100) == 47

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghizyxwvutsrqponmlkjihgfedcba", k = 100) == 35

Answer: assert Solution().longestPalindromicSubsequence(s = "deified", k = 2) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaabbbbccccdddd", k = 10) == 13

Answer: assert Solution().longestPalindromicSubsequence(s = "madam", k = 2) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "rotor", k = 3) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefgihgfedcb", k = 15) == 15

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghihgfedcba", k = 10) == 17

Answer: assert Solution().longestPalindromicSubsequence(s = "pqrstuvwxyzabcdefghijklmno", k = 50) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", k = 0) == 30

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyz", k = 10) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxzy", k = 100) == 26

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzz", k = 10) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 150) == 49

Answer: assert Solution().longestPalindromicSubsequence(s = "abcabcabcabcabcabcabcabcabcabc", k = 100) == 30

Answer: assert Solution().longestPalindromicSubsequence(s = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", k = 5) == 52

Answer: assert Solution().longestPalindromicSubsequence(s = "reviled", k = 7) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "abccba", k = 0) == 6

Answer: assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyx", k = 100) == 21

Answer: assert Solution().longestPalindromicSubsequence(s = "pneumonoultramicroscopicsilicovolcanoconiosis", k = 50) == 37

Answer: assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyz", k = 10) == 6

Answer: assert Solution().longestPalindromicSubsequence(s = "step on no pets", k = 10) == 15

Answer: assert Solution().longestPalindromicSubsequence(s = "zyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyxzyx", k = 100) == 42

Answer: assert Solution().longestPalindromicSubsequence(s = "abababababababababababababababab", k = 50) == 32

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyz", k = 50) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "mississippi", k = 3) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "abacabadabacaba", k = 5) == 15

Answer: assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzzyxwvutsrqponmlk", k = 80) == 30

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdezyxwvutsrqponmlkjihgf", k = 50) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "abacabadabacaba", k = 10) == 15

Answer: assert Solution().longestPalindromicSubsequence(s = "abcabcabc", k = 4) == 8

Answer: assert Solution().longestPalindromicSubsequence(s = "abababababababababab", k = 10) == 20

Answer: assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 100) == 40

Answer: assert Solution().longestPalindromicSubsequence(s = "abcabcabcabcabcabcabc", k = 20) == 21

Answer: assert Solution().longestPalindromicSubsequence(s = "qwertyuiopasdfghjklzxcvbnm", k = 25) == 18

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaaabbbbb", k = 10) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "zabacabadabacaba", k = 10) == 15

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefedcbaabcdefedcbaabcdefedcba", k = 60) == 33

Answer: assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 150) == 36

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdxyzzyxcba", k = 10) == 13

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 100) == 51

Answer: assert Solution().longestPalindromicSubsequence(s = "kayak", k = 20) == 5

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyz", k = 100) == 40

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdzdcba", k = 3) == 9

Answer: assert Solution().longestPalindromicSubsequence(s = "aababaaa", k = 2) == 8

Answer: assert Solution().longestPalindromicSubsequence(s = "deed", k = 3) == 4

Answer: assert Solution().longestPalindromicSubsequence(s = "ababababababababababababababababababababababababababab", k = 100) == 54

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijkmlkjihgfedcba", k = 50) == 24

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdeedcba", k = 3) == 10

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefg", k = 10) == 6

Answer: assert Solution().longestPalindromicSubsequence(s = "mnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 50) == 30

Answer: assert Solution().longestPalindromicSubsequence(s = "racecar", k = 0) == 7

Answer: assert Solution().longestPalindromicSubsequence(s = "zyxwvutsrqponmlkjihgfedcba", k = 150) == 26

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba", k = 50) == 51

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijabcdefghij", k = 15) == 13

Answer: assert Solution().longestPalindromicSubsequence(s = "abacabadabacabadabacaba", k = 20) == 23

Answer: assert Solution().longestPalindromicSubsequence(s = "aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa", k = 5) == 52

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdabcdabcdabcd", k = 20) == 16

Answer: assert Solution().longestPalindromicSubsequence(s = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", k = 180) == 52

Answer: assert Solution().longestPalindromicSubsequence(s = "abcdefghijjihgfedcba", k = 5) == 20

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 longestPalindromicSubsequence(self, s: str, k: int) -> int:
        @cache
        def dfs(i: int, j: int, k: int) -> int:
            if i > j:
                return 0
            if i == j:
                return 1
            res = max(dfs(i + 1, j, k), dfs(i, j - 1, k))
            d = abs(s[i] - s[j])
            t = min(d, 26 - d)
            if t <= k:
                res = max(res, dfs(i + 1, j - 1, k - t) + 2)
            return res

        s = list(map(ord, s))
        n = len(s)
        ans = dfs(0, n - 1, k)
        dfs.cache_clear()
        return ans

class Solution:
    def longestPalindromicSubsequence(self, s: str, k: int) -> int: