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 caption of length n. A good caption is a string where every character appears in groups of at least 3 consecutive occurrences.
For example:

"aaabbb" and "aaaaccc" are good captions.
"aabbb" and "ccccd" are not good captions.

You can perform the following operation any number of times:
Choose an index i (where 0 <= i < n) and change the character at that index to either:

The character immediately before it in the alphabet (if caption[i] != 'a').
The character immediately after it in the alphabet (if caption[i] != 'z').

Your task is to convert the given caption into a good caption using the minimum number of operations, and return it. If there are multiple possible good captions, return the lexicographically smallest one among them. If it is impossible to create a good caption, return an empty string "".

Example 1:

Input: caption = "cdcd"
Output: "cccc"
Explanation:
It can be shown that the given caption cannot be transformed into a good caption with fewer than 2 operations. The possible good captions that can be created using exactly 2 operations are:

"dddd": Change caption[0] and caption[2] to their next character 'd'.
"cccc": Change caption[1] and caption[3] to their previous character 'c'.

Since "cccc" is lexicographically smaller than "dddd", return "cccc".

Example 2:

Input: caption = "aca"
Output: "aaa"
Explanation:
It can be proven that the given caption requires at least 2 operations to be transformed into a good caption. The only good caption that can be obtained with exactly 2 operations is as follows:

Operation 1: Change caption[1] to 'b'. caption = "aba".
Operation 2: Change caption[1] to 'a'. caption = "aaa".

Thus, return "aaa".

Example 3:

Input: caption = "bc"
Output: ""
Explanation:
It can be shown that the given caption cannot be converted to a good caption by using any number of operations.

Constraints:

1 <= caption.length <= 5 * 104
caption consists only of lowercase English letters.

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

SOLUTION:

Question:
PROBLEM:
You are given a string caption of length n. A good caption is a string where every character appears in groups of at least 3 consecutive occurrences.
For example:

"aaabbb" and "aaaaccc" are good captions.
"aabbb" and "ccccd" are not good captions.

You can perform the following operation any number of times:
Choose an index i (where 0 <= i < n) and change the character at that index to either:

The character immediately before it in the alphabet (if caption[i] != 'a').
The character immediately after it in the alphabet (if caption[i] != 'z').

Your task is to convert the given caption into a good caption using the minimum number of operations, and return it. If there are multiple possible good captions, return the lexicographically smallest one among them. If it is impossible to create a good caption, return an empty string "".

Example 1:

Input: caption = "cdcd"
Output: "cccc"
Explanation:
It can be shown that the given caption cannot be transformed into a good caption with fewer than 2 operations. The possible good captions that can be created using exactly 2 operations are:

"dddd": Change caption[0] and caption[2] to their next character 'd'.
"cccc": Change caption[1] and caption[3] to their previous character 'c'.

Since "cccc" is lexicographically smaller than "dddd", return "cccc".

Example 2:

Input: caption = "aca"
Output: "aaa"
Explanation:
It can be proven that the given caption requires at least 2 operations to be transformed into a good caption. The only good caption that can be obtained with exactly 2 operations is as follows:

Operation 1: Change caption[1] to 'b'. caption = "aba".
Operation 2: Change caption[1] to 'a'. caption = "aaa".

Thus, return "aaa".

Example 3:

Input: caption = "bc"
Output: ""
Explanation:
It can be shown that the given caption cannot be converted to a good caption by using any number of operations.

Constraints:

1 <= caption.length <= 5 * 104
caption consists only of lowercase English letters.

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

SOLUTION:

assert Solution().minCostGoodCaption(caption = "cdcd") == "cccc"

assert Solution().minCostGoodCaption(caption = "xyz") == "yyy"

assert Solution().minCostGoodCaption(caption = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz") == "aaaaccceeefffhhhiiikkklllnnnoooqqqrrrtttuuuwwwxxxzzz"

assert Solution().minCostGoodCaption(caption = "aabbbccc") == "bbbbbccc"

assert Solution().minCostGoodCaption(caption = "abcdefghijklmnopqrstuvwxyz") == "bbbbffffjjjmmmpppsssvvvyyy"

assert Solution().minCostGoodCaption(caption = "cccccc") == "cccccc"

assert Solution().minCostGoodCaption(caption = "abcdefg") == "bbbbfff"

assert Solution().minCostGoodCaption(caption = "aaabbbccc") == "aaabbbccc"

assert Solution().minCostGoodCaption(caption = "aaa") == "aaa"

assert Solution().minCostGoodCaption(caption = "aabbaa") == "aaaaaa"

assert Solution().minCostGoodCaption(caption = "abca") == "aaaa"

assert Solution().minCostGoodCaption(caption = "aabbcc") == "aaaccc"

assert Solution().minCostGoodCaption(caption = "zzz") == "zzz"

assert Solution().minCostGoodCaption(caption = "zzzzzzzzz") == "zzzzzzzzz"

assert Solution().minCostGoodCaption(caption = "abacaba") == "aaaaaaa"

assert Solution().minCostGoodCaption(caption = "aaabbb") == "aaabbb"

assert Solution().minCostGoodCaption(caption = "aca") == "aaa"

assert Solution().minCostGoodCaption(caption = "bc") == ""

assert Solution().minCostGoodCaption(caption = "abcabcabc") == "aaaabbbbb"

assert Solution().minCostGoodCaption(caption = "cccbbbaaaa") == "cccbbbaaaa"

assert Solution().minCostGoodCaption(caption = "abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyz") == "bbbbffffjjjmmmpppsssvvvyyybbbbffffjjjmmmpppsssvvvyyy"

assert Solution().minCostGoodCaption(caption = "abcdefabcfedcba") == "bbbeeebbbeeebbb"

assert Solution().minCostGoodCaption(caption = "bcdfghjklmnpqrstvwxyzbcdfghjklmnpqrstvwxyz") == "cccgggkkknnnrrrvvvyyycccgggkkknnnrrrvvvyyy"

assert Solution().minCostGoodCaption(caption = "abcdcba") == "bbbbbbb"

assert Solution().minCostGoodCaption(caption = "aaaaabbbbbcccccc") == "aaaaabbbbbcccccc"

assert Solution().minCostGoodCaption(caption = "abcdabcdabcd") == "bbbbbbcccccc"

assert Solution().minCostGoodCaption(caption = "qweqweqwe") == "qqqqqeeee"

assert Solution().minCostGoodCaption(caption = "abcabcabcabcabcabcabcabc") == "aaaabbbbbaaaabbbbbbbbbbb"

assert Solution().minCostGoodCaption(caption = "mnopqrstuvw") == "nnnnrrrrvvv"

assert Solution().minCostGoodCaption(caption = "aaaabbbaaaa") == "aaaabbbaaaa"

assert Solution().minCostGoodCaption(caption = "zzzyzzyzz") == "zzzzzzzzz"

assert Solution().minCostGoodCaption(caption = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba") == "bbbeeehhhkkknnnqqqtttwwwyyywwwtttqqqnnnkkkhhheeebbb"

assert Solution().minCostGoodCaption(caption = "mnopqr") == "nnnqqq"

assert Solution().minCostGoodCaption(caption = "aaaabbbbcccc") == "aaaabbbbcccc"

assert Solution().minCostGoodCaption(caption = "aaabbbccccddddeeeeffff") == "aaabbbccccddddeeeeffff"

assert Solution().minCostGoodCaption(caption = "aaaabbaaa") == "aaabbbaaa"

assert Solution().minCostGoodCaption(caption = "xyzxyzxyz") == "xxxxyyyyy"

assert Solution().minCostGoodCaption(caption = "abcdeedcba") == "bbbddddbbb"

assert Solution().minCostGoodCaption(caption = "zyxzyxzyxzyxzyx") == "yyyyyxxxxyyyyyy"

assert Solution().minCostGoodCaption(caption = "aaaazzzzzzzzzzzz") == "aaaazzzzzzzzzzzz"

assert Solution().minCostGoodCaption(caption = "xyzzzzzzzzzzzzzyx") == "yyyzzzzzzzzzzzyyy"

assert Solution().minCostGoodCaption(caption = "aabbaabbaabb") == "aaaaaabbbbbb"

assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz"

assert Solution().minCostGoodCaption(caption = "aabbccddeeff") == "aaacccdddfff"

assert Solution().minCostGoodCaption(caption = "abcdefghi") == "bbbeeehhh"

assert Solution().minCostGoodCaption(caption = "aaaabbbbccccdddd") == "aaaabbbbccccdddd"

assert Solution().minCostGoodCaption(caption = "mnopqrs") == "nnnnrrr"

assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz"

assert Solution().minCostGoodCaption(caption = "ababababab") == "aaaaaaabbb"

assert Solution().minCostGoodCaption(caption = "yzabcyzabcyzabc") == "yyycccbbbyyybbb"

assert Solution().minCostGoodCaption(caption = "zzzzzzyyy") == "zzzzzzyyy"

assert Solution().minCostGoodCaption(caption = "zyxwvutsrqponmlkjihgfedcba") == "xxxxttttqqqnnnkkkhhheeebbb"

assert Solution().minCostGoodCaption(caption = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzz") == "aaaacccceeeggghhhjjjkkkmmmnnnpppqqqssstttvvvwwwyyyzzz"

assert Solution().minCostGoodCaption(caption = "mnopqrstuvmnopqrstuvmnopqrstuvmnopqrstuvmnopqrstuvmnopqrstuv") == "nnnnrrruuunnnnrrruuunnnnrrruuunnnnrrruuunnnnrrruuunnnnrrruuu"

assert Solution().minCostGoodCaption(caption = "aaazzz") == "aaazzz"

assert Solution().minCostGoodCaption(caption = "fedcbazyxwvutsrqponmlkjihgfedcba") == "eeebbbxxxxttttqqqnnnkkkhhheeebbb"

assert Solution().minCostGoodCaption(caption = "aabbaabbcc") == "aaaaaaaccc"

assert Solution().minCostGoodCaption(caption = "xyzyxzyxzyxzyx") == "yyyyxxxxyyyyyy"

assert Solution().minCostGoodCaption(caption = "abcdefghijk") == "bbbbffffjjj"

assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzz"

assert Solution().minCostGoodCaption(caption = "zzzzzyyyy") == "zzzzzyyyy"

assert Solution().minCostGoodCaption(caption = "xyzzxyzzxyzz") == "yyyyyyzzzzzz"

assert Solution().minCostGoodCaption(caption = "xyzabc") == "yyybbb"

assert Solution().minCostGoodCaption(caption = "aabccccaaa") == "aaaccccaaa"

assert Solution().minCostGoodCaption(caption = "aaaaaaaaaaa") == "aaaaaaaaaaa"

assert Solution().minCostGoodCaption(caption = "mnopqrstu") == "nnnqqqttt"

assert Solution().minCostGoodCaption(caption = "abcdefgabcdefgabcdefg") == "bbbbfffbbbbfffbbbbfff"

assert Solution().minCostGoodCaption(caption = "zzzzzyyyyy") == "zzzzzyyyyy"

assert Solution().minCostGoodCaption(caption = "mnopqrsmnopqrs") == "nnnnrrrnnnnrrr"

assert Solution().minCostGoodCaption(caption = "zzzzz") == "zzzzz"

assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzaaa") == "zzzzzzzzzzzzaaa"

assert Solution().minCostGoodCaption(caption = "zzzaaa") == "zzzaaa"

assert Solution().minCostGoodCaption(caption = "abcdabcde") == "bbbbbbddd"

Answer: assert Solution().minCostGoodCaption(caption = "cdcd") == "cccc"

Answer: assert Solution().minCostGoodCaption(caption = "xyz") == "yyy"

Answer: assert Solution().minCostGoodCaption(caption = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz") == "aaaaccceeefffhhhiiikkklllnnnoooqqqrrrtttuuuwwwxxxzzz"

Answer: assert Solution().minCostGoodCaption(caption = "aabbbccc") == "bbbbbccc"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefghijklmnopqrstuvwxyz") == "bbbbffffjjjmmmpppsssvvvyyy"

Answer: assert Solution().minCostGoodCaption(caption = "cccccc") == "cccccc"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefg") == "bbbbfff"

Answer: assert Solution().minCostGoodCaption(caption = "aaabbbccc") == "aaabbbccc"

Answer: assert Solution().minCostGoodCaption(caption = "aaa") == "aaa"

Answer: assert Solution().minCostGoodCaption(caption = "aabbaa") == "aaaaaa"

Answer: assert Solution().minCostGoodCaption(caption = "abca") == "aaaa"

Answer: assert Solution().minCostGoodCaption(caption = "aabbcc") == "aaaccc"

Answer: assert Solution().minCostGoodCaption(caption = "zzz") == "zzz"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzzzz") == "zzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "abacaba") == "aaaaaaa"

Answer: assert Solution().minCostGoodCaption(caption = "aaabbb") == "aaabbb"

Answer: assert Solution().minCostGoodCaption(caption = "aca") == "aaa"

Answer: assert Solution().minCostGoodCaption(caption = "bc") == ""

Answer: assert Solution().minCostGoodCaption(caption = "abcabcabc") == "aaaabbbbb"

Answer: assert Solution().minCostGoodCaption(caption = "cccbbbaaaa") == "cccbbbaaaa"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefghijklmnopqrstuvwxyzabcdefghijklmnopqrstuvwxyz") == "bbbbffffjjjmmmpppsssvvvyyybbbbffffjjjmmmpppsssvvvyyy"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefabcfedcba") == "bbbeeebbbeeebbb"

Answer: assert Solution().minCostGoodCaption(caption = "bcdfghjklmnpqrstvwxyzbcdfghjklmnpqrstvwxyz") == "cccgggkkknnnrrrvvvyyycccgggkkknnnrrrvvvyyy"

Answer: assert Solution().minCostGoodCaption(caption = "abcdcba") == "bbbbbbb"

Answer: assert Solution().minCostGoodCaption(caption = "aaaaabbbbbcccccc") == "aaaaabbbbbcccccc"

Answer: assert Solution().minCostGoodCaption(caption = "abcdabcdabcd") == "bbbbbbcccccc"

Answer: assert Solution().minCostGoodCaption(caption = "qweqweqwe") == "qqqqqeeee"

Answer: assert Solution().minCostGoodCaption(caption = "abcabcabcabcabcabcabcabc") == "aaaabbbbbaaaabbbbbbbbbbb"

Answer: assert Solution().minCostGoodCaption(caption = "mnopqrstuvw") == "nnnnrrrrvvv"

Answer: assert Solution().minCostGoodCaption(caption = "aaaabbbaaaa") == "aaaabbbaaaa"

Answer: assert Solution().minCostGoodCaption(caption = "zzzyzzyzz") == "zzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefghijklmnopqrstuvwxyzyxwvutsrqponmlkjihgfedcba") == "bbbeeehhhkkknnnqqqtttwwwyyywwwtttqqqnnnkkkhhheeebbb"

Answer: assert Solution().minCostGoodCaption(caption = "mnopqr") == "nnnqqq"

Answer: assert Solution().minCostGoodCaption(caption = "aaaabbbbcccc") == "aaaabbbbcccc"

Answer: assert Solution().minCostGoodCaption(caption = "aaabbbccccddddeeeeffff") == "aaabbbccccddddeeeeffff"

Answer: assert Solution().minCostGoodCaption(caption = "aaaabbaaa") == "aaabbbaaa"

Answer: assert Solution().minCostGoodCaption(caption = "xyzxyzxyz") == "xxxxyyyyy"

Answer: assert Solution().minCostGoodCaption(caption = "abcdeedcba") == "bbbddddbbb"

Answer: assert Solution().minCostGoodCaption(caption = "zyxzyxzyxzyxzyx") == "yyyyyxxxxyyyyyy"

Answer: assert Solution().minCostGoodCaption(caption = "aaaazzzzzzzzzzzz") == "aaaazzzzzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "xyzzzzzzzzzzzzzyx") == "yyyzzzzzzzzzzzyyy"

Answer: assert Solution().minCostGoodCaption(caption = "aabbaabbaabb") == "aaaaaabbbbbb"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "aabbccddeeff") == "aaacccdddfff"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefghi") == "bbbeeehhh"

Answer: assert Solution().minCostGoodCaption(caption = "aaaabbbbccccdddd") == "aaaabbbbccccdddd"

Answer: assert Solution().minCostGoodCaption(caption = "mnopqrs") == "nnnnrrr"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "ababababab") == "aaaaaaabbb"

Answer: assert Solution().minCostGoodCaption(caption = "yzabcyzabcyzabc") == "yyycccbbbyyybbb"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzyyy") == "zzzzzzyyy"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "zyxwvutsrqponmlkjihgfedcba") == "xxxxttttqqqnnnkkkhhheeebbb"

Answer: assert Solution().minCostGoodCaption(caption = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzz") == "aaaacccceeeggghhhjjjkkkmmmnnnpppqqqssstttvvvwwwyyyzzz"

Answer: assert Solution().minCostGoodCaption(caption = "mnopqrstuvmnopqrstuvmnopqrstuvmnopqrstuvmnopqrstuvmnopqrstuv") == "nnnnrrruuunnnnrrruuunnnnrrruuunnnnrrruuunnnnrrruuunnnnrrruuu"

Answer: assert Solution().minCostGoodCaption(caption = "aaazzz") == "aaazzz"

Answer: assert Solution().minCostGoodCaption(caption = "fedcbazyxwvutsrqponmlkjihgfedcba") == "eeebbbxxxxttttqqqnnnkkkhhheeebbb"

Answer: assert Solution().minCostGoodCaption(caption = "aabbaabbcc") == "aaaaaaaccc"

Answer: assert Solution().minCostGoodCaption(caption = "xyzyxzyxzyxzyx") == "yyyyxxxxyyyyyy"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefghijk") == "bbbbffffjjj"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzyyyy") == "zzzzzyyyy"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz") == "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "xyzzxyzzxyzz") == "yyyyyyzzzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "xyzabc") == "yyybbb"

Answer: assert Solution().minCostGoodCaption(caption = "aabccccaaa") == "aaaccccaaa"

Answer: assert Solution().minCostGoodCaption(caption = "aaaaaaaaaaa") == "aaaaaaaaaaa"

Answer: assert Solution().minCostGoodCaption(caption = "mnopqrstu") == "nnnqqqttt"

Answer: assert Solution().minCostGoodCaption(caption = "aaaabbbbccccddddeeeeffffgggghhhhiiiijjjjkkkkllllmmmmnnnnooooppppqqqqrrrrssssttttuuuuvvvvwwwwxxxxyyyyzzzz") == "aaaabbbbccccddddeeeeffffgggghhhhiiiijjjjkkkkllllmmmmnnnnooooppppqqqqrrrrssssttttuuuuvvvvwwwwxxxxyyyyzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "abcdefgabcdefgabcdefg") == "bbbbfffbbbbfffbbbbfff"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzyyyyy") == "zzzzzyyyyy"

Answer: assert Solution().minCostGoodCaption(caption = "mnopqrsmnopqrs") == "nnnnrrrnnnnrrr"

Answer: assert Solution().minCostGoodCaption(caption = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzzaabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz") == "aaaaccceeefffhhhiiikkklllnnnoooqqqrrrtttuuuwwwxxxzzzaaaaccceeefffhhhiiikkklllnnnoooqqqrrrtttuuuwwwxxxzzz"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzz") == "zzzzz"

Answer: assert Solution().minCostGoodCaption(caption = "zzzzzzzzzzzzaaa") == "zzzzzzzzzzzzaaa"

Answer: assert Solution().minCostGoodCaption(caption = "zzzaaa") == "zzzaaa"

Answer: assert Solution().minCostGoodCaption(caption = "abcdabcde") == "bbbbbbddd"

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 minCostGoodCaption(self, caption: str) -> str:
    n = len(caption)
    if n < 3:
      return ''

    MAX_COST = 1_000_000_000
    # dp[i][j][k] := the minimum cost of caption[i..n - 1], where j is the last
    # letter used, and k is the count of consecutive letters
    dp = [[[MAX_COST] * 3 for _ in range(26)] for _ in range(n)]

    for c in range(26):
      dp[-1][c][0] = abs(string.ascii_lowercase.index(caption[-1]) - c)

    minCost = MAX_COST

    for i in range(n - 2, -1, -1):
      newMinCost = MAX_COST
      for c in range(26):
        changeCost = abs(string.ascii_lowercase.index(caption[i]) - c)
        dp[i][c][0] = changeCost + minCost
        dp[i][c][1] = changeCost + dp[i + 1][c][0]
        dp[i][c][2] = changeCost + min(dp[i + 1][c][1], dp[i + 1][c][2])
        newMinCost = min(newMinCost, dp[i][c][2])
      minCost = newMinCost

    # Reconstruct the string.
    ans = []
    cost = MAX_COST
    letter = -1

    # Find the initial best letter.
    for c in range(25, -1, -1):
      if dp[0][c][2] <= cost:
        letter = c
        cost = dp[0][c][2]

    # Add the initial triplet.
    cost -= self._appendLetter(caption, 0, chr(ord('a') + letter), ans)
    cost -= self._appendLetter(caption, 1, chr(ord('a') + letter), ans)
    cost -= self._appendLetter(caption, 2, chr(ord('a') + letter), ans)

    # Build the rest of the string.
    i = 3
    while i < n:
      nextLetter = self._getNextLetter(dp, i, cost)
      if nextLetter < letter or min(dp[i][letter]) > cost:
        letter = nextLetter
        cost -= self._appendLetter(caption, i, chr(ord('a') + letter), ans)
        cost -= self._appendLetter(caption, i + 1, chr(ord('a') + letter), ans)
        cost -= self._appendLetter(caption, i + 2, chr(ord('a') + letter), ans)
        i += 3
      else:
        cost -= self._appendLetter(caption, i, chr(ord('a') + letter), ans)
        i += 1

    return ''.join(ans)

  def _getNextLetter(self, dp: list[list[list[int]]], i: int, cost: int) -> int:
    nextLetter = 26
    for c in range(25, -1, -1):
      if cost == dp[i][c][2]:
        nextLetter = c
    return nextLetter

  def _appendLetter(
      self,
      caption: str,
      i: int,
      letter: str,
      ans: list[str]
  ) -> int:
    ans.append(letter)
    return abs(ord(caption[i]) - ord(letter))

class Solution:
    def minCostGoodCaption(self, caption: str) -> str:


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 consisting of the characters 'N', 'S', 'E', and 'W', where s[i] indicates movements in an infinite grid:

'N' : Move north by 1 unit.
'S' : Move south by 1 unit.
'E' : Move east by 1 unit.
'W' : Move west by 1 unit.

Initially, you are at the origin (0, 0). You can change at most k characters to any of the four directions.
Find the maximum Manhattan distance from the origin that can be achieved at any time while performing the movements in order.
The Manhattan Distance between two cells (xi, yi) and (xj, yj) is |xi - xj| + |yi - yj|.

Example 1:

Input: s = "NWSE", k = 1
Output: 3
Explanation:
Change s[2] from 'S' to 'N'. The string s becomes "NWNE".

Movement
Position (x, y)
Manhattan Distance
Maximum

s[0] == 'N'
(0, 1)
0 + 1 = 1
1

s[1] == 'W'
(-1, 1)
1 + 1 = 2
2

s[2] == 'N'
(-1, 2)
1 + 2 = 3
3

s[3] == 'E'
(0, 2)
0 + 2 = 2
3

The maximum Manhattan distance from the origin that can be achieved is 3. Hence, 3 is the output.

Example 2:

Input: s = "NSWWEW", k = 3
Output: 6
Explanation:
Change s[1] from 'S' to 'N', and s[4] from 'E' to 'W'. The string s becomes "NNWWWW".
The maximum Manhattan distance from the origin that can be achieved is 6. Hence, 6 is the output.

Constraints:

1 <= s.length <= 105
0 <= k <= s.length
s consists of only 'N', 'S', 'E', and 'W'.

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

SOLUTION:

Question:
PROBLEM:
You are given a string s consisting of the characters 'N', 'S', 'E', and 'W', where s[i] indicates movements in an infinite grid:

'N' : Move north by 1 unit.
'S' : Move south by 1 unit.
'E' : Move east by 1 unit.
'W' : Move west by 1 unit.

Initially, you are at the origin (0, 0). You can change at most k characters to any of the four directions.
Find the maximum Manhattan distance from the origin that can be achieved at any time while performing the movements in order.
The Manhattan Distance between two cells (xi, yi) and (xj, yj) is |xi - xj| + |yi - yj|.

Example 1:

Input: s = "NWSE", k = 1
Output: 3
Explanation:
Change s[2] from 'S' to 'N'. The string s becomes "NWNE".

Movement
Position (x, y)
Manhattan Distance
Maximum

s[0] == 'N'
(0, 1)
0 + 1 = 1
1

s[1] == 'W'
(-1, 1)
1 + 1 = 2
2

s[2] == 'N'
(-1, 2)
1 + 2 = 3
3

s[3] == 'E'
(0, 2)
0 + 2 = 2
3

The maximum Manhattan distance from the origin that can be achieved is 3. Hence, 3 is the output.

Example 2:

Input: s = "NSWWEW", k = 3
Output: 6
Explanation:
Change s[1] from 'S' to 'N', and s[4] from 'E' to 'W'. The string s becomes "NNWWWW".
The maximum Manhattan distance from the origin that can be achieved is 6. Hence, 6 is the output.

Constraints:

1 <= s.length <= 105
0 <= k <= s.length
s consists of only 'N', 'S', 'E', and 'W'.

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

SOLUTION:

assert Solution().maxDistance(s = "NSWWEW", k = 3) == 6

assert Solution().maxDistance(s = "EEEEEE", k = 0) == 6

assert Solution().maxDistance(s = "NNN", k = 0) == 3

assert Solution().maxDistance(s = "NWSE", k = 1) == 3

assert Solution().maxDistance(s = "EEEE", k = 4) == 4

assert Solution().maxDistance(s = "NNNN", k = 0) == 4

assert Solution().maxDistance(s = "SSSSS", k = 5) == 5

assert Solution().maxDistance(s = "NNNSSS", k = 3) == 6

assert Solution().maxDistance(s = "WEWEWEW", k = 1) == 3

assert Solution().maxDistance(s = "NSEW", k = 4) == 4

assert Solution().maxDistance(s = "NENENE", k = 2) == 6

assert Solution().maxDistance(s = "EEEEEE", k = 2) == 6

assert Solution().maxDistance(s = "NESW", k = 2) == 4

assert Solution().maxDistance(s = "NESW", k = 0) == 2

assert Solution().maxDistance(s = "NNNSSS", k = 2) == 5

assert Solution().maxDistance(s = "NSEW", k = 0) == 1

assert Solution().maxDistance(s = "WWWW", k = 4) == 4

assert Solution().maxDistance(s = "NNSSEW", k = 2) == 5

assert Solution().maxDistance(s = "EEEEE", k = 3) == 5

assert Solution().maxDistance(s = "EWEWWE", k = 0) == 1

assert Solution().maxDistance(s = "NEENSWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWW", k = 25) == 83

assert Solution().maxDistance(s = "EWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEW", k = 100) == 124

assert Solution().maxDistance(s = "NNSSSSSSNNSSSSSS", k = 5) == 16

assert Solution().maxDistance(s = "NNSWWSWWSSSWSSS", k = 6) == 15

assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEE", k = 15) == 19

assert Solution().maxDistance(s = "NNNSSSWWEEEEE", k = 12) == 13

assert Solution().maxDistance(s = "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", k = 20) == 38

assert Solution().maxDistance(s = "WSSSNNNEEEWWWEWWWSSEENEEEEEWWEWWEWEEWWWW", k = 15) == 34

assert Solution().maxDistance(s = "NNNSSSSEEEEEWWWWWW", k = 5) == 14

assert Solution().maxDistance(s = "SSSSSSSSSS", k = 10) == 10

assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 10) == 108

assert Solution().maxDistance(s = "NNNEEESWWWWW", k = 5) == 12

assert Solution().maxDistance(s = "EEEEWWWWEEEEWWWWEEEEWWWW", k = 20) == 24

assert Solution().maxDistance(s = "EWWWNNEEESSSSS", k = 7) == 14

assert Solution().maxDistance(s = "NESWNESWNESWNESWNESW", k = 10) == 20

assert Solution().maxDistance(s = "NNNNNSSSSS", k = 5) == 10

assert Solution().maxDistance(s = "EEEEEEEEEEWWWWWWWWWW", k = 10) == 20

assert Solution().maxDistance(s = "EEEEEEEEEEEEEEE", k = 7) == 15

assert Solution().maxDistance(s = "NNNSSSSSSSSSSEEEEEEEWWWWWWWWWWWWWW", k = 15) == 34

assert Solution().maxDistance(s = "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", k = 20) == 118

assert Solution().maxDistance(s = "NENEWENEWENEWENEWENEWENEWENEWENEWENEWENEWEN", k = 12) == 43

assert Solution().maxDistance(s = "NESESWNWSWNENWSESNWENS", k = 7) == 15

assert Solution().maxDistance(s = "E", k = 1) == 1

assert Solution().maxDistance(s = "NESEWESWESWNWNEN", k = 12) == 16

assert Solution().maxDistance(s = "NENWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNW", k = 30) == 116

assert Solution().maxDistance(s = "WWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWW", k = 10) == 57

assert Solution().maxDistance(s = "NENWNWNWSENWSW", k = 4) == 14

assert Solution().maxDistance(s = "NENENENENENENENENEN", k = 10) == 19

assert Solution().maxDistance(s = "NSEWNNSEWNNSEWNNSEWNNSEWNNSEWNNSEW", k = 15) == 34

assert Solution().maxDistance(s = "WNNNWWNNEEENSSS", k = 7) == 15

assert Solution().maxDistance(s = "NESWNESWNESWNESWNESWNESWNESWNESWNESWNESWNESW", k = 10) == 22

assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSSSSS", k = 10) == 19

assert Solution().maxDistance(s = "WWSWNWNWNWWSWNWSWWSWNWNW", k = 15) == 24

assert Solution().maxDistance(s = "NENENENENENENENENENENE", k = 20) == 22

assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 10) == 36

assert Solution().maxDistance(s = "NESENESWNWSWNENE", k = 7) == 16

assert Solution().maxDistance(s = "NNNNSSSSSSSSEEEE", k = 10) == 16

assert Solution().maxDistance(s = "EWSWNESWNWESWNESWNWESWNESWNWESWNESWNWESWNWESW", k = 15) == 37

assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 25) == 49

assert Solution().maxDistance(s = "WNWWSWEESSNNNSSWWS", k = 12) == 18

assert Solution().maxDistance(s = "NENWSWESWNENWSWESWNENWSW", k = 15) == 24

assert Solution().maxDistance(s = "WSSSWNWEWEW", k = 5) == 11

assert Solution().maxDistance(s = "WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW", k = 30) == 50

assert Solution().maxDistance(s = "SWNESWSESWSWSEWSEWSEWSEWSEWSEWSEWSEWSEWSEWS", k = 25) == 43

assert Solution().maxDistance(s = "NNNSSSSSSSSSSSSSSSSS", k = 5) == 20

assert Solution().maxDistance(s = "SSSSWWWWNNNNEEEE", k = 10) == 16

assert Solution().maxDistance(s = "SSSSSSSSSSEEEEEEEEEEWWWWWWWWWWWNNNNNNNNNNN", k = 20) == 42

assert Solution().maxDistance(s = "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", k = 50) == 46

assert Solution().maxDistance(s = "NSEWNESEWNESEWNESEWNESEW", k = 15) == 24

assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS", k = 30) == 40

assert Solution().maxDistance(s = "NNNSSWSSWEEWWNNNSSW", k = 10) == 19

assert Solution().maxDistance(s = "EEWWEWNEWEWNENW", k = 10) == 15

assert Solution().maxDistance(s = "SSSSSSSSSSNNNNNNNNNNEEEEEEEEEEWWWWWWWWWW", k = 10) == 30

assert Solution().maxDistance(s = "NSWWWNWWWNWSW", k = 5) == 13

assert Solution().maxDistance(s = "NNNWWWEESSEENNNWWEESSEENNNWWEESSEENNNWWEESSEENNNWWEESSEENNNWWEESSEENNN", k = 25) == 70

assert Solution().maxDistance(s = "WNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNW", k = 50) == 125

assert Solution().maxDistance(s = "EESSWWNNNNNSSSWEWEWNWNWSSNENENESWEWEWWSWWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWSWWSWSW", k = 50) == 120

assert Solution().maxDistance(s = "NNSSEEEWW", k = 10) == 9

assert Solution().maxDistance(s = "NENEWNWNENEW", k = 6) == 12

assert Solution().maxDistance(s = "NNNNEEEEWWWWSSSSNNNNEEEEWWWWSSSS", k = 20) == 32

assert Solution().maxDistance(s = "WWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWW", k = 20) == 44

assert Solution().maxDistance(s = "SWWNSWEWNEWNNSWEN", k = 6) == 17

assert Solution().maxDistance(s = "NNNNNSSSSSNEEWWW", k = 3) == 9

assert Solution().maxDistance(s = "EEENSSSSSNWWSWEEWEESSSNWWSWEEW", k = 10) == 29

assert Solution().maxDistance(s = "NNSWNNWSSWNNWSSW", k = 8) == 16

assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS", k = 15) == 114

assert Solution().maxDistance(s = "SWWNSESNWNENWNWS", k = 8) == 16

assert Solution().maxDistance(s = "NSEWNESEWNESEW", k = 8) == 14

assert Solution().maxDistance(s = "NSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSS", k = 15) == 44

assert Solution().maxDistance(s = "WSSSWWSSSWWSSSWWSSSWW", k = 15) == 21

assert Solution().maxDistance(s = "EESSWWNN", k = 4) == 8

assert Solution().maxDistance(s = "SSSSNNNNSSSSNNNN", k = 8) == 16

assert Solution().maxDistance(s = "NSSSNSEWEEW", k = 5) == 11

assert Solution().maxDistance(s = "NSEWSEWNSEW", k = 2) == 6

assert Solution().maxDistance(s = "SSSSNNNNWWWWEEEEWWWWWWWWWSSSSSSSSSNNNNNNNNNNEEEEEEEEENNNNNNNNNNWWWWWWWWWEEEEEEEEESSSSSSSSSSSSSNNNNNNNNNNNNWWWWWWWW", k = 25) == 70

assert Solution().maxDistance(s = "NNNNNNNNNNNSSSSSSSSSSSSSSSSSSSSSSSSSSS", k = 15) == 38

assert Solution().maxDistance(s = "NWNWNWNWNWNSNSNSNS", k = 7) == 18

assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSS", k = 10) == 16

assert Solution().maxDistance(s = "EWNESWNWNENEWNEWWNWNWNE", k = 15) == 23

assert Solution().maxDistance(s = "N", k = 0) == 1

assert Solution().maxDistance(s = "WENWENWENWENWENWENWENWENWENWENWENWENWENWENWE", k = 30) == 44

assert Solution().maxDistance(s = "NEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 10) == 43

assert Solution().maxDistance(s = "EWEWNEWNENW", k = 7) == 11

assert Solution().maxDistance(s = "SNENWNWNWNENEWSNWNWNWNWNWNWNWNW", k = 12) == 31

assert Solution().maxDistance(s = "NNNNWWWWSSSSEEEEWWWWSSSSEEEEWWWWSSSSEEEE", k = 25) == 40

assert Solution().maxDistance(s = "NNSWWNWWSWNESEWEEWSSSNNNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWES", k = 40) == 103

assert Solution().maxDistance(s = "NENENENENENENENENENENENENENENENENENENENENENENENENENENENENENE", k = 20) == 60

assert Solution().maxDistance(s = "NNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEE", k = 30) == 62

Answer: assert Solution().maxDistance(s = "NSWWEW", k = 3) == 6

Answer: assert Solution().maxDistance(s = "EEEEEE", k = 0) == 6

Answer: assert Solution().maxDistance(s = "NNN", k = 0) == 3

Answer: assert Solution().maxDistance(s = "NWSE", k = 1) == 3

Answer: assert Solution().maxDistance(s = "EEEE", k = 4) == 4

Answer: assert Solution().maxDistance(s = "NNNN", k = 0) == 4

Answer: assert Solution().maxDistance(s = "SSSSS", k = 5) == 5

Answer: assert Solution().maxDistance(s = "NNNSSS", k = 3) == 6

Answer: assert Solution().maxDistance(s = "WEWEWEW", k = 1) == 3

Answer: assert Solution().maxDistance(s = "NSEW", k = 4) == 4

Answer: assert Solution().maxDistance(s = "NENENE", k = 2) == 6

Answer: assert Solution().maxDistance(s = "EEEEEE", k = 2) == 6

Answer: assert Solution().maxDistance(s = "NESW", k = 2) == 4

Answer: assert Solution().maxDistance(s = "NESW", k = 0) == 2

Answer: assert Solution().maxDistance(s = "NNNSSS", k = 2) == 5

Answer: assert Solution().maxDistance(s = "NSEW", k = 0) == 1

Answer: assert Solution().maxDistance(s = "WWWW", k = 4) == 4

Answer: assert Solution().maxDistance(s = "NNSSEW", k = 2) == 5

Answer: assert Solution().maxDistance(s = "EEEEE", k = 3) == 5

Answer: assert Solution().maxDistance(s = "EWEWWE", k = 0) == 1

Answer: assert Solution().maxDistance(s = "NEENSWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWW", k = 25) == 83

Answer: assert Solution().maxDistance(s = "EWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEWEW", k = 100) == 124

Answer: assert Solution().maxDistance(s = "NNSSSSSSNNSSSSSS", k = 5) == 16

Answer: assert Solution().maxDistance(s = "NNSWWSWWSSSWSSS", k = 6) == 15

Answer: assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEE", k = 15) == 19

Answer: assert Solution().maxDistance(s = "NNNSSSWWEEEEE", k = 12) == 13

Answer: assert Solution().maxDistance(s = "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", k = 20) == 38

Answer: assert Solution().maxDistance(s = "WSSSNNNEEEWWWEWWWSSEENEEEEEWWEWWEWEEWWWW", k = 15) == 34

Answer: assert Solution().maxDistance(s = "NNNSSSSEEEEEWWWWWW", k = 5) == 14

Answer: assert Solution().maxDistance(s = "SSSSSSSSSS", k = 10) == 10

Answer: assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 10) == 108

Answer: assert Solution().maxDistance(s = "NNNEEESWWWWW", k = 5) == 12

Answer: assert Solution().maxDistance(s = "EEEEWWWWEEEEWWWWEEEEWWWW", k = 20) == 24

Answer: assert Solution().maxDistance(s = "EWWWNNEEESSSSS", k = 7) == 14

Answer: assert Solution().maxDistance(s = "NESWNESWNESWNESWNESW", k = 10) == 20

Answer: assert Solution().maxDistance(s = "NNNNNSSSSS", k = 5) == 10

Answer: assert Solution().maxDistance(s = "EEEEEEEEEEWWWWWWWWWW", k = 10) == 20

Answer: assert Solution().maxDistance(s = "EEEEEEEEEEEEEEE", k = 7) == 15

Answer: assert Solution().maxDistance(s = "NNNSSSSSSSSSSEEEEEEEWWWWWWWWWWWWWW", k = 15) == 34

Answer: assert Solution().maxDistance(s = "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", k = 20) == 118

Answer: assert Solution().maxDistance(s = "NENEWENEWENEWENEWENEWENEWENEWENEWENEWENEWEN", k = 12) == 43

Answer: assert Solution().maxDistance(s = "NESESWNWSWNENWSESNWENS", k = 7) == 15

Answer: assert Solution().maxDistance(s = "E", k = 1) == 1

Answer: assert Solution().maxDistance(s = "NESEWESWESWNWNEN", k = 12) == 16

Answer: assert Solution().maxDistance(s = "NENWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNWENWNWNWSWNWNW", k = 30) == 116

Answer: assert Solution().maxDistance(s = "WWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWWEWW", k = 10) == 57

Answer: assert Solution().maxDistance(s = "NENWNWNWSENWSW", k = 4) == 14

Answer: assert Solution().maxDistance(s = "NENENENENENENENENEN", k = 10) == 19

Answer: assert Solution().maxDistance(s = "NSEWNNSEWNNSEWNNSEWNNSEWNNSEWNNSEW", k = 15) == 34

Answer: assert Solution().maxDistance(s = "WNNNWWNNEEENSSS", k = 7) == 15

Answer: assert Solution().maxDistance(s = "NESWNESWNESWNESWNESWNESWNESWNESWNESWNESWNESW", k = 10) == 22

Answer: assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSSSSS", k = 10) == 19

Answer: assert Solution().maxDistance(s = "WWSWNWNWNWWSWNWSWWSWNWNW", k = 15) == 24

Answer: assert Solution().maxDistance(s = "NENENENENENENENENENENE", k = 20) == 22

Answer: assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 10) == 36

Answer: assert Solution().maxDistance(s = "NESENESWNWSWNENE", k = 7) == 16

Answer: assert Solution().maxDistance(s = "NNNNSSSSSSSSEEEE", k = 10) == 16

Answer: assert Solution().maxDistance(s = "EWSWNESWNWESWNESWNWESWNESWNWESWNESWNWESWNWESW", k = 15) == 37

Answer: assert Solution().maxDistance(s = "EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 25) == 49

Answer: assert Solution().maxDistance(s = "WNWWSWEESSNNNSSWWS", k = 12) == 18

Answer: assert Solution().maxDistance(s = "NENWSWESWNENWSWESWNENWSW", k = 15) == 24

Answer: assert Solution().maxDistance(s = "WSSSWNWEWEW", k = 5) == 11

Answer: assert Solution().maxDistance(s = "WWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWWW", k = 30) == 50

Answer: assert Solution().maxDistance(s = "SWNESWSESWSWSEWSEWSEWSEWSEWSEWSEWSEWSEWSEWS", k = 25) == 43

Answer: assert Solution().maxDistance(s = "NNNSSSSSSSSSSSSSSSSS", k = 5) == 20

Answer: assert Solution().maxDistance(s = "SSSSWWWWNNNNEEEE", k = 10) == 16

Answer: assert Solution().maxDistance(s = "SSSSSSSSSSEEEEEEEEEEWWWWWWWWWWWNNNNNNNNNNN", k = 20) == 42

Answer: assert Solution().maxDistance(s = "NNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNNN", k = 50) == 46

Answer: assert Solution().maxDistance(s = "NSEWNESEWNESEWNESEWNESEW", k = 15) == 24

Answer: assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS", k = 30) == 40

Answer: assert Solution().maxDistance(s = "NNNSSWSSWEEWWNNNSSW", k = 10) == 19

Answer: assert Solution().maxDistance(s = "EEWWEWNEWEWNENW", k = 10) == 15

Answer: assert Solution().maxDistance(s = "SSSSSSSSSSNNNNNNNNNNEEEEEEEEEEWWWWWWWWWW", k = 10) == 30

Answer: assert Solution().maxDistance(s = "NSWWWNWWWNWSW", k = 5) == 13

Answer: assert Solution().maxDistance(s = "NNNWWWEESSEENNNWWEESSEENNNWWEESSEENNNWWEESSEENNNWWEESSEENNNWWEESSEENNN", k = 25) == 70

Answer: assert Solution().maxDistance(s = "WNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNWNW", k = 50) == 125

Answer: assert Solution().maxDistance(s = "EESSWWNNNNNSSSWEWEWNWNWSSNENENESWEWEWWSWWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWWSWSWSWWSWSW", k = 50) == 120

Answer: assert Solution().maxDistance(s = "NNSSEEEWW", k = 10) == 9

Answer: assert Solution().maxDistance(s = "NENEWNWNENEW", k = 6) == 12

Answer: assert Solution().maxDistance(s = "NNNNEEEEWWWWSSSSNNNNEEEEWWWWSSSS", k = 20) == 32

Answer: assert Solution().maxDistance(s = "WWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWWSWW", k = 20) == 44

Answer: assert Solution().maxDistance(s = "SWWNSWEWNEWNNSWEN", k = 6) == 17

Answer: assert Solution().maxDistance(s = "NNNNNSSSSSNEEWWW", k = 3) == 9

Answer: assert Solution().maxDistance(s = "EEENSSSSSNWWSWEEWEESSSNWWSWEEW", k = 10) == 29

Answer: assert Solution().maxDistance(s = "NNSWNNWSSWNNWSSW", k = 8) == 16

Answer: assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS", k = 15) == 114

Answer: assert Solution().maxDistance(s = "SWWNSESNWNENWNWS", k = 8) == 16

Answer: assert Solution().maxDistance(s = "NSEWNESEWNESEW", k = 8) == 14

Answer: assert Solution().maxDistance(s = "NSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSSNSSS", k = 15) == 44

Answer: assert Solution().maxDistance(s = "WSSSWWSSSWWSSSWWSSSWW", k = 15) == 21

Answer: assert Solution().maxDistance(s = "EESSWWNN", k = 4) == 8

Answer: assert Solution().maxDistance(s = "SSSSNNNNSSSSNNNN", k = 8) == 16

Answer: assert Solution().maxDistance(s = "NSSSNSEWEEW", k = 5) == 11

Answer: assert Solution().maxDistance(s = "NSEWSEWNSEW", k = 2) == 6

Answer: assert Solution().maxDistance(s = "SSSSNNNNWWWWEEEEWWWWWWWWWSSSSSSSSSNNNNNNNNNNEEEEEEEEENNNNNNNNNNWWWWWWWWWEEEEEEEEESSSSSSSSSSSSSNNNNNNNNNNNNWWWWWWWW", k = 25) == 70

Answer: assert Solution().maxDistance(s = "NNNNNNNNNNNSSSSSSSSSSSSSSSSSSSSSSSSSSS", k = 15) == 38

Answer: assert Solution().maxDistance(s = "NWNWNWNWNWNSNSNSNS", k = 7) == 18

Answer: assert Solution().maxDistance(s = "SSSSSSSSSSSSSSSS", k = 10) == 16

Answer: assert Solution().maxDistance(s = "EWNESWNWNENEWNEWWNWNWNE", k = 15) == 23

Answer: assert Solution().maxDistance(s = "N", k = 0) == 1

Answer: assert Solution().maxDistance(s = "WENWENWENWENWENWENWENWENWENWENWENWENWENWENWE", k = 30) == 44

Answer: assert Solution().maxDistance(s = "NEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE", k = 10) == 43

Answer: assert Solution().maxDistance(s = "EWEWNEWNENW", k = 7) == 11

Answer: assert Solution().maxDistance(s = "SNENWNWNWNENEWSNWNWNWNWNWNWNWNW", k = 12) == 31

Answer: assert Solution().maxDistance(s = "NNNNWWWWSSSSEEEEWWWWSSSSEEEEWWWWSSSSEEEE", k = 25) == 40

Answer: assert Solution().maxDistance(s = "NNSWWNWWSWNESEWEEWSSSNNNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWESWNWES", k = 40) == 103

Answer: assert Solution().maxDistance(s = "NENENENENENENENENENENENENENENENENENENENENENENENENENENENENENE", k = 20) == 60

Answer: assert Solution().maxDistance(s = "NNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEEWWNNSSEE", k = 30) == 62

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 maxDistance(self, s: str, k: int) -> int:
        def calc(a: str, b: str) -> int:
            ans = mx = cnt = 0
            for c in s:
                if c == a or c == b:
                    mx += 1
                elif cnt < k:
                    cnt += 1
                    mx += 1
                else:
                    mx -= 1
                ans = max(ans, mx)
            return ans

        a = calc("S", "E")
        b = calc("S", "W")
        c = calc("N", "E")
        d = calc("N", "W")
        return max(a, b, c, d)

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


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

PROBLEM:
You are given two arrays, nums and target.
In a single operation, you may increment any element of nums by 1.
Return the minimum number of operations required so that each element in target has at least one multiple in nums.

Example 1:

Input: nums = [1,2,3], target = [4]
Output: 1
Explanation:
The minimum number of operations required to satisfy the condition is 1.

Increment 3 to 4 with just one operation, making 4 a multiple of itself.

Example 2:

Input: nums = [8,4], target = [10,5]
Output: 2
Explanation:
The minimum number of operations required to satisfy the condition is 2.

Increment 8 to 10 with 2 operations, making 10 a multiple of both 5 and 10.

Example 3:

Input: nums = [7,9,10], target = [7]
Output: 0
Explanation:
Target 7 already has a multiple in nums, so no additional operations are needed.

Constraints:

1 <= nums.length <= 5 * 104
1 <= target.length <= 4
target.length <= nums.length
1 <= nums[i], target[i] <= 104

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

SOLUTION:

Question:
PROBLEM:
You are given two arrays, nums and target.
In a single operation, you may increment any element of nums by 1.
Return the minimum number of operations required so that each element in target has at least one multiple in nums.

Example 1:

Input: nums = [1,2,3], target = [4]
Output: 1
Explanation:
The minimum number of operations required to satisfy the condition is 1.

Increment 3 to 4 with just one operation, making 4 a multiple of itself.

Example 2:

Input: nums = [8,4], target = [10,5]
Output: 2
Explanation:
The minimum number of operations required to satisfy the condition is 2.

Increment 8 to 10 with 2 operations, making 10 a multiple of both 5 and 10.

Example 3:

Input: nums = [7,9,10], target = [7]
Output: 0
Explanation:
Target 7 already has a multiple in nums, so no additional operations are needed.

Constraints:

1 <= nums.length <= 5 * 104
1 <= target.length <= 4
target.length <= nums.length
1 <= nums[i], target[i] <= 104

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

SOLUTION:

assert Solution().minimumIncrements(nums = [5,10,15], target = [5,10,15]) == 0

assert Solution().minimumIncrements(nums = [5,10,15,20], target = [5,15]) == 0

assert Solution().minimumIncrements(nums = [10,20,30], target = [5,15]) == 0

assert Solution().minimumIncrements(nums = [1,1,1], target = [3]) == 2

assert Solution().minimumIncrements(nums = [10,20,30,40], target = [5,10,15]) == 0

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

assert Solution().minimumIncrements(nums = [100,200,300], target = [100,200,300]) == 0

assert Solution().minimumIncrements(nums = [100,200,300], target = [50,100]) == 0

assert Solution().minimumIncrements(nums = [5,10,15], target = [5,10]) == 0

assert Solution().minimumIncrements(nums = [3,6,9,12], target = [3,6]) == 0

assert Solution().minimumIncrements(nums = [7,9,10], target = [7]) == 0

assert Solution().minimumIncrements(nums = [1,5,10], target = [2,5,10]) == 0

assert Solution().minimumIncrements(nums = [2,3,5], target = [6,10]) == 8

assert Solution().minimumIncrements(nums = [15,20,25], target = [5,10]) == 0

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

assert Solution().minimumIncrements(nums = [1,1,1], target = [2,3,4]) == 5

assert Solution().minimumIncrements(nums = [8,4], target = [10,5]) == 2

assert Solution().minimumIncrements(nums = [2,4,6,8], target = [2,4]) == 0

assert Solution().minimumIncrements(nums = [5,10,15], target = [25]) == 10

assert Solution().minimumIncrements(nums = [2,4,6,8], target = [1,2,4,8]) == 0

assert Solution().minimumIncrements(nums = [1,3,5,7], target = [2,4,6]) == 2

assert Solution().minimumIncrements(nums = [1,1,1,1], target = [1]) == 0

assert Solution().minimumIncrements(nums = [3,6,9], target = [3,6,9]) == 0

assert Solution().minimumIncrements(nums = [3,6,9,12], target = [3,6,9]) == 0

assert Solution().minimumIncrements(nums = [10,20,30,40], target = [10,20,30]) == 0

assert Solution().minimumIncrements(nums = [2,2,2,2], target = [2,4]) == 2

assert Solution().minimumIncrements(nums = [13, 26, 39, 52], target = [26, 52]) == 0

assert Solution().minimumIncrements(nums = [6, 12, 18, 24, 30], target = [24, 30]) == 0

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

assert Solution().minimumIncrements(nums = [1,3,5,7,9], target = [2,4,6,8]) == 2

assert Solution().minimumIncrements(nums = [33, 66, 99, 132, 165], target = [11, 33]) == 0

assert Solution().minimumIncrements(nums = [3, 9, 27, 81], target = [81]) == 0

assert Solution().minimumIncrements(nums = [8,16,24,32,40,48], target = [9,18,27,36]) == 7

assert Solution().minimumIncrements(nums = [5,10,15,20,25,30], target = [12,24,36]) == 10

assert Solution().minimumIncrements(nums = [1, 1, 1, 1, 1], target = [2, 3, 5]) == 7

assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11, 13, 17], target = [21, 29]) == 20

assert Solution().minimumIncrements(nums = [11, 22, 33, 44, 55], target = [33, 44, 55]) == 0

assert Solution().minimumIncrements(nums = [2, 4, 6, 8, 10], target = [12, 15]) == 9

assert Solution().minimumIncrements(nums = [100,200,300,400,500], target = [250,500]) == 0

assert Solution().minimumIncrements(nums = [2, 5, 7, 11, 13, 17, 19], target = [10, 14, 21]) == 6

assert Solution().minimumIncrements(nums = [11, 22, 33, 44, 55], target = [22, 44]) == 0

assert Solution().minimumIncrements(nums = [3,6,9,12,15], target = [18,30]) == 21

assert Solution().minimumIncrements(nums = [6, 7, 8, 9, 10], target = [15, 20]) == 16

assert Solution().minimumIncrements(nums = [2,4,8,16,32], target = [5,10,20]) == 4

assert Solution().minimumIncrements(nums = [7, 14, 21, 28], target = [14, 28]) == 0

assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35], target = [28, 35]) == 0

assert Solution().minimumIncrements(nums = [8, 16, 24, 32, 40], target = [16, 32]) == 0

assert Solution().minimumIncrements(nums = [11,22,33,44,55], target = [5,10,20]) == 5

assert Solution().minimumIncrements(nums = [100, 200, 300, 400], target = [200, 300]) == 0

assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35], target = [35, 42]) == 14

assert Solution().minimumIncrements(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target = [20, 25]) == 26

assert Solution().minimumIncrements(nums = [5,15,25,35,45], target = [25,50,75]) == 45

assert Solution().minimumIncrements(nums = [11,22,33,44,55], target = [55,33]) == 0

assert Solution().minimumIncrements(nums = [2,3,5,7], target = [14,21,35]) == 55

assert Solution().minimumIncrements(nums = [3, 3, 3, 3, 3], target = [3, 6]) == 3

assert Solution().minimumIncrements(nums = [50, 100, 150, 200], target = [105, 175]) == 30

assert Solution().minimumIncrements(nums = [13,26,39,52,65], target = [26,52,78]) == 13

assert Solution().minimumIncrements(nums = [100, 200, 300], target = [50, 150]) == 0

assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25, 30], target = [30, 40]) == 15

assert Solution().minimumIncrements(nums = [19, 38, 57, 76], target = [38, 76]) == 0

assert Solution().minimumIncrements(nums = [4, 8, 12, 16, 20], target = [20, 24]) == 8

assert Solution().minimumIncrements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], target = [3, 5, 7]) == 12

assert Solution().minimumIncrements(nums = [3,9,27,81,243], target = [1,3,9,27,81]) == 0

assert Solution().minimumIncrements(nums = [14, 28, 42, 56, 70], target = [56, 70]) == 0

assert Solution().minimumIncrements(nums = [12,24,36,48], target = [36,48]) == 0

assert Solution().minimumIncrements(nums = [7,14,21,28,35], target = [3,6,9]) == 1

assert Solution().minimumIncrements(nums = [1, 3, 5, 7, 9, 11], target = [9, 18]) == 7

assert Solution().minimumIncrements(nums = [12,24,36,48,60], target = [13,26]) == 2

assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11], target = [30, 70]) == 82

assert Solution().minimumIncrements(nums = [2,4,6,8,10], target = [15,25]) == 22

assert Solution().minimumIncrements(nums = [9, 18, 27, 36], target = [18, 36]) == 0

assert Solution().minimumIncrements(nums = [4, 8, 12, 16, 20], target = [16, 24, 32]) == 20

assert Solution().minimumIncrements(nums = [5,15,25,35,45], target = [35,45]) == 0

assert Solution().minimumIncrements(nums = [11, 13, 17, 19], target = [12, 14, 18]) == 3

assert Solution().minimumIncrements(nums = [11, 22, 33, 44, 55], target = [44, 55]) == 0

assert Solution().minimumIncrements(nums = [5,10,15,20,25,30], target = [11,22]) == 2

assert Solution().minimumIncrements(nums = [12, 15, 18, 21], target = [6, 9]) == 0

assert Solution().minimumIncrements(nums = [3, 6, 9, 12], target = [15, 18]) == 12

assert Solution().minimumIncrements(nums = [1,2,4,8,16,32], target = [16,8]) == 0

assert Solution().minimumIncrements(nums = [12, 15, 18, 20], target = [6, 10]) == 0

assert Solution().minimumIncrements(nums = [100, 200, 300], target = [150, 300]) == 0

assert Solution().minimumIncrements(nums = [50, 100, 150, 200], target = [100, 150, 200]) == 0

assert Solution().minimumIncrements(nums = [13,26,39,52,65], target = [39,65]) == 0

assert Solution().minimumIncrements(nums = [9, 18, 27, 36, 45], target = [27, 36]) == 0

assert Solution().minimumIncrements(nums = [9, 18, 27, 36, 45], target = [18, 27, 36]) == 0

assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25], target = [25, 50]) == 25

assert Solution().minimumIncrements(nums = [12, 24, 36, 48, 60], target = [28, 42, 56]) == 14

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

assert Solution().minimumIncrements(nums = [3, 6, 9, 12, 15, 18], target = [9, 15]) == 0

assert Solution().minimumIncrements(nums = [3, 9, 15, 21, 27], target = [21, 27]) == 0

assert Solution().minimumIncrements(nums = [8,16,24,32,40,48], target = [48,24]) == 0

assert Solution().minimumIncrements(nums = [1, 10, 100, 1000], target = [1000]) == 0

assert Solution().minimumIncrements(nums = [100,200,300,400], target = [300,200,100]) == 0

assert Solution().minimumIncrements(nums = [6,12,18,24,30], target = [36,72,108]) == 126

assert Solution().minimumIncrements(nums = [7,14,21,28,35], target = [3,9,27]) == 6

assert Solution().minimumIncrements(nums = [17, 34, 51, 68, 85], target = [17, 34, 51]) == 0

assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25], target = [25, 30]) == 10

assert Solution().minimumIncrements(nums = [4, 8, 12, 16], target = [8, 12]) == 0

assert Solution().minimumIncrements(nums = [10, 20, 30, 40, 50], target = [20, 30, 40, 50]) == 0

assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25], target = [20, 30]) == 5

assert Solution().minimumIncrements(nums = [20, 40, 60, 80, 100], target = [40, 80]) == 0

assert Solution().minimumIncrements(nums = [10,20,30,40,50], target = [6,12,18]) == 6

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

assert Solution().minimumIncrements(nums = [3,6,9,12,15], target = [18,21]) == 12

assert Solution().minimumIncrements(nums = [15, 30, 45, 60], target = [30, 60]) == 0

assert Solution().minimumIncrements(nums = [3,6,9,12,15], target = [18,9]) == 3

assert Solution().minimumIncrements(nums = [1, 5, 9, 13, 17, 21], target = [6, 12, 18]) == 4

assert Solution().minimumIncrements(nums = [7,14,21,28], target = [21,14,7]) == 0

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

assert Solution().minimumIncrements(nums = [12, 18, 24, 30], target = [15, 20]) == 2

assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11], target = [21, 28, 35]) == 61

assert Solution().minimumIncrements(nums = [7,14,21,28,35], target = [28,35]) == 0

assert Solution().minimumIncrements(nums = [8,16,24,32,40], target = [16,32,48]) == 8

assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35], target = [14, 28, 42]) == 7

assert Solution().minimumIncrements(nums = [4, 8, 12, 16], target = [9, 15]) == 4

assert Solution().minimumIncrements(nums = [3, 6, 9, 12, 15], target = [18, 21]) == 12

assert Solution().minimumIncrements(nums = [2, 5, 8, 11, 14], target = [11, 22]) == 8

assert Solution().minimumIncrements(nums = [17, 34, 51, 68], target = [68, 85]) == 34

assert Solution().minimumIncrements(nums = [12, 24, 36, 48, 60], target = [48, 60]) == 0

assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11], target = [6, 10]) == 4

assert Solution().minimumIncrements(nums = [5,10,15,20,25,30], target = [25,35]) == 5

assert Solution().minimumIncrements(nums = [11, 22, 33, 44], target = [33, 44]) == 0

assert Solution().minimumIncrements(nums = [13, 26, 39, 52, 65], target = [52, 65]) == 0

Answer: assert Solution().minimumIncrements(nums = [5,10,15], target = [5,10,15]) == 0

Answer: assert Solution().minimumIncrements(nums = [5,10,15,20], target = [5,15]) == 0

Answer: assert Solution().minimumIncrements(nums = [10,20,30], target = [5,15]) == 0

Answer: assert Solution().minimumIncrements(nums = [1,1,1], target = [3]) == 2

Answer: assert Solution().minimumIncrements(nums = [10,20,30,40], target = [5,10,15]) == 0

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

Answer: assert Solution().minimumIncrements(nums = [100,200,300], target = [100,200,300]) == 0

Answer: assert Solution().minimumIncrements(nums = [100,200,300], target = [50,100]) == 0

Answer: assert Solution().minimumIncrements(nums = [5,10,15], target = [5,10]) == 0

Answer: assert Solution().minimumIncrements(nums = [3,6,9,12], target = [3,6]) == 0

Answer: assert Solution().minimumIncrements(nums = [7,9,10], target = [7]) == 0

Answer: assert Solution().minimumIncrements(nums = [1,5,10], target = [2,5,10]) == 0

Answer: assert Solution().minimumIncrements(nums = [2,3,5], target = [6,10]) == 8

Answer: assert Solution().minimumIncrements(nums = [15,20,25], target = [5,10]) == 0

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

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

Answer: assert Solution().minimumIncrements(nums = [8,4], target = [10,5]) == 2

Answer: assert Solution().minimumIncrements(nums = [2,4,6,8], target = [2,4]) == 0

Answer: assert Solution().minimumIncrements(nums = [5,10,15], target = [25]) == 10

Answer: assert Solution().minimumIncrements(nums = [2,4,6,8], target = [1,2,4,8]) == 0

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

Answer: assert Solution().minimumIncrements(nums = [1,1,1,1], target = [1]) == 0

Answer: assert Solution().minimumIncrements(nums = [3,6,9], target = [3,6,9]) == 0

Answer: assert Solution().minimumIncrements(nums = [3,6,9,12], target = [3,6,9]) == 0

Answer: assert Solution().minimumIncrements(nums = [10,20,30,40], target = [10,20,30]) == 0

Answer: assert Solution().minimumIncrements(nums = [2,2,2,2], target = [2,4]) == 2

Answer: assert Solution().minimumIncrements(nums = [13, 26, 39, 52], target = [26, 52]) == 0

Answer: assert Solution().minimumIncrements(nums = [6, 12, 18, 24, 30], target = [24, 30]) == 0

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

Answer: assert Solution().minimumIncrements(nums = [1,3,5,7,9,11,13,15,17,19], target = [2,4,6,8]) == 2

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

Answer: assert Solution().minimumIncrements(nums = [9,18,27,36,45,54,63,72,81,90], target = [27,54,81]) == 0

Answer: assert Solution().minimumIncrements(nums = [33, 66, 99, 132, 165], target = [11, 33]) == 0

Answer: assert Solution().minimumIncrements(nums = [3, 9, 27, 81], target = [81]) == 0

Answer: assert Solution().minimumIncrements(nums = [4,8,12,16,20,24,28,32,36,40], target = [24,36,48]) == 8

Answer: assert Solution().minimumIncrements(nums = [8,16,24,32,40,48], target = [9,18,27,36]) == 7

Answer: assert Solution().minimumIncrements(nums = [5,10,15,20,25,30], target = [12,24,36]) == 10

Answer: assert Solution().minimumIncrements(nums = [1, 1, 1, 1, 1], target = [2, 3, 5]) == 7

Answer: assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11, 13, 17], target = [21, 29]) == 20

Answer: assert Solution().minimumIncrements(nums = [11, 22, 33, 44, 55], target = [33, 44, 55]) == 0

Answer: assert Solution().minimumIncrements(nums = [2, 4, 6, 8, 10], target = [12, 15]) == 9

Answer: assert Solution().minimumIncrements(nums = [100,200,300,400,500], target = [250,500]) == 0

Answer: assert Solution().minimumIncrements(nums = [2,6,10,14,18,22,26,30,34,38,42], target = [4,8,12,16]) == 4

Answer: assert Solution().minimumIncrements(nums = [2, 5, 7, 11, 13, 17, 19], target = [10, 14, 21]) == 6

Answer: assert Solution().minimumIncrements(nums = [11, 22, 33, 44, 55], target = [22, 44]) == 0

Answer: assert Solution().minimumIncrements(nums = [3,6,9,12,15], target = [18,30]) == 21

Answer: assert Solution().minimumIncrements(nums = [6, 7, 8, 9, 10], target = [15, 20]) == 16

Answer: assert Solution().minimumIncrements(nums = [2,4,8,16,32], target = [5,10,20]) == 4

Answer: assert Solution().minimumIncrements(nums = [7, 14, 21, 28], target = [14, 28]) == 0

Answer: assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35], target = [28, 35]) == 0

Answer: assert Solution().minimumIncrements(nums = [8, 16, 24, 32, 40], target = [16, 32]) == 0

Answer: assert Solution().minimumIncrements(nums = [11,22,33,44,55], target = [5,10,20]) == 5

Answer: assert Solution().minimumIncrements(nums = [100, 200, 300, 400], target = [200, 300]) == 0

Answer: assert Solution().minimumIncrements(nums = [25, 50, 75, 100, 125], target = [50, 100, 150]) == 25

Answer: assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35], target = [35, 42]) == 14

Answer: assert Solution().minimumIncrements(nums = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], target = [20, 25]) == 26

Answer: assert Solution().minimumIncrements(nums = [5,15,25,35,45], target = [25,50,75]) == 45

Answer: assert Solution().minimumIncrements(nums = [11,22,33,44,55], target = [55,33]) == 0

Answer: assert Solution().minimumIncrements(nums = [2,3,5,7], target = [14,21,35]) == 55

Answer: assert Solution().minimumIncrements(nums = [3, 3, 3, 3, 3], target = [3, 6]) == 3

Answer: assert Solution().minimumIncrements(nums = [50, 100, 150, 200], target = [105, 175]) == 30

Answer: assert Solution().minimumIncrements(nums = [100,200,300,400], target = [125,250,375]) == 125

Answer: assert Solution().minimumIncrements(nums = [13,26,39,52,65], target = [26,52,78]) == 13

Answer: assert Solution().minimumIncrements(nums = [100, 200, 300], target = [50, 150]) == 0

Answer: assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25, 30], target = [30, 40]) == 15

Answer: assert Solution().minimumIncrements(nums = [19, 38, 57, 76], target = [38, 76]) == 0

Answer: assert Solution().minimumIncrements(nums = [4, 8, 12, 16, 20], target = [20, 24]) == 8

Answer: assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35, 42, 49, 56, 63, 70], target = [5, 10, 15, 20]) == 4

Answer: assert Solution().minimumIncrements(nums = [1, 1, 1, 1, 1, 1, 1, 1, 1, 1], target = [3, 5, 7]) == 12

Answer: assert Solution().minimumIncrements(nums = [3,9,27,81,243], target = [1,3,9,27,81]) == 0

Answer: assert Solution().minimumIncrements(nums = [14, 28, 42, 56, 70], target = [56, 70]) == 0

Answer: assert Solution().minimumIncrements(nums = [12,24,36,48], target = [36,48]) == 0

Answer: assert Solution().minimumIncrements(nums = [10, 20, 30, 40, 50, 60, 70, 80, 90, 100], target = [9, 18, 27, 36]) == 3

Answer: assert Solution().minimumIncrements(nums = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], target = [10, 20, 30]) == 12

Answer: assert Solution().minimumIncrements(nums = [7,14,21,28,35], target = [3,6,9]) == 1

Answer: assert Solution().minimumIncrements(nums = [1, 3, 5, 7, 9, 11], target = [9, 18]) == 7

Answer: assert Solution().minimumIncrements(nums = [5, 15, 25, 35, 45, 55, 65, 75, 85, 95], target = [2, 3, 7, 11]) == 1

Answer: assert Solution().minimumIncrements(nums = [12,24,36,48,60], target = [13,26]) == 2

Answer: assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11], target = [30, 70]) == 82

Answer: assert Solution().minimumIncrements(nums = [2,4,6,8,10], target = [15,25]) == 22

Answer: assert Solution().minimumIncrements(nums = [9, 18, 27, 36], target = [18, 36]) == 0

Answer: assert Solution().minimumIncrements(nums = [4, 8, 12, 16, 20], target = [16, 24, 32]) == 20

Answer: assert Solution().minimumIncrements(nums = [5,15,25,35,45], target = [35,45]) == 0

Answer: assert Solution().minimumIncrements(nums = [11, 13, 17, 19], target = [12, 14, 18]) == 3

Answer: assert Solution().minimumIncrements(nums = [11, 22, 33, 44, 55], target = [44, 55]) == 0

Answer: assert Solution().minimumIncrements(nums = [5,10,15,20,25,30], target = [11,22]) == 2

Answer: assert Solution().minimumIncrements(nums = [17, 19, 23, 29, 31, 37, 41, 43, 47, 53], target = [18, 20, 21, 22]) == 4

Answer: assert Solution().minimumIncrements(nums = [12, 15, 18, 21], target = [6, 9]) == 0

Answer: assert Solution().minimumIncrements(nums = [3, 6, 9, 12, 15, 18, 21, 24, 27, 30], target = [21, 24, 27]) == 0

Answer: assert Solution().minimumIncrements(nums = [3, 6, 9, 12], target = [15, 18]) == 12

Answer: assert Solution().minimumIncrements(nums = [3,6,9,12,15,18,21,24,27], target = [18,27,36]) == 12

Answer: assert Solution().minimumIncrements(nums = [1,2,4,8,16,32], target = [16,8]) == 0

Answer: assert Solution().minimumIncrements(nums = [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], target = [3, 7, 11]) == 1

Answer: assert Solution().minimumIncrements(nums = [12, 15, 18, 20], target = [6, 10]) == 0

Answer: assert Solution().minimumIncrements(nums = [100, 200, 300], target = [150, 300]) == 0

Answer: assert Solution().minimumIncrements(nums = [2,3,5,7,11,13,17,19], target = [10,15,21]) == 7

Answer: assert Solution().minimumIncrements(nums = [50, 100, 150, 200], target = [100, 150, 200]) == 0

Answer: assert Solution().minimumIncrements(nums = [13,26,39,52,65], target = [39,65]) == 0

Answer: assert Solution().minimumIncrements(nums = [9, 18, 27, 36, 45], target = [27, 36]) == 0

Answer: assert Solution().minimumIncrements(nums = [9, 18, 27, 36, 45], target = [18, 27, 36]) == 0

Answer: assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25], target = [25, 50]) == 25

Answer: assert Solution().minimumIncrements(nums = [12, 24, 36, 48, 60], target = [28, 42, 56]) == 14

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

Answer: assert Solution().minimumIncrements(nums = [4, 8, 12, 16, 20, 24, 28, 32, 36, 40], target = [6, 14, 21]) == 1

Answer: assert Solution().minimumIncrements(nums = [3, 6, 9, 12, 15, 18], target = [9, 15]) == 0

Answer: assert Solution().minimumIncrements(nums = [3, 9, 15, 21, 27], target = [21, 27]) == 0

Answer: assert Solution().minimumIncrements(nums = [13, 26, 39, 52, 65, 78, 91, 104, 117, 130], target = [4, 8, 12, 16]) == 4

Answer: assert Solution().minimumIncrements(nums = [8,16,24,32,40,48], target = [48,24]) == 0

Answer: assert Solution().minimumIncrements(nums = [1, 10, 100, 1000], target = [1000]) == 0

Answer: assert Solution().minimumIncrements(nums = [1000,2000,3000,4000], target = [500,1500,2500]) == 500

Answer: assert Solution().minimumIncrements(nums = [100,200,300,400], target = [300,200,100]) == 0

Answer: assert Solution().minimumIncrements(nums = [6,12,18,24,30], target = [36,72,108]) == 126

Answer: assert Solution().minimumIncrements(nums = [7,14,21,28,35], target = [3,9,27]) == 6

Answer: assert Solution().minimumIncrements(nums = [17, 34, 51, 68, 85], target = [17, 34, 51]) == 0

Answer: assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25], target = [25, 30]) == 10

Answer: assert Solution().minimumIncrements(nums = [4, 8, 12, 16], target = [8, 12]) == 0

Answer: assert Solution().minimumIncrements(nums = [10, 20, 30, 40, 50], target = [20, 30, 40, 50]) == 0

Answer: assert Solution().minimumIncrements(nums = [5, 10, 15, 20, 25], target = [20, 30]) == 5

Answer: assert Solution().minimumIncrements(nums = [20, 40, 60, 80, 100], target = [40, 80]) == 0

Answer: assert Solution().minimumIncrements(nums = [10,20,30,40,50], target = [6,12,18]) == 6

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

Answer: assert Solution().minimumIncrements(nums = [3, 6, 9, 12, 15, 18, 21, 24, 27, 30], target = [5, 10, 15]) == 0

Answer: assert Solution().minimumIncrements(nums = [3,6,9,12,15], target = [18,21]) == 12

Answer: assert Solution().minimumIncrements(nums = [15, 30, 45, 60], target = [30, 60]) == 0

Answer: assert Solution().minimumIncrements(nums = [3,6,9,12,15], target = [18,9]) == 3

Answer: assert Solution().minimumIncrements(nums = [1000, 2000, 3000, 4000, 5000], target = [1500, 2500]) == 0

Answer: assert Solution().minimumIncrements(nums = [1, 5, 9, 13, 17, 21], target = [6, 12, 18]) == 4

Answer: assert Solution().minimumIncrements(nums = [7,14,21,28], target = [21,14,7]) == 0

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

Answer: assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35, 42, 49, 56, 63, 70], target = [14, 21, 28]) == 0

Answer: assert Solution().minimumIncrements(nums = [12, 18, 24, 30], target = [15, 20]) == 2

Answer: assert Solution().minimumIncrements(nums = [1,1,1,1,1,1,1,1,1,1], target = [5,10,15]) == 23

Answer: assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11], target = [21, 28, 35]) == 61

Answer: assert Solution().minimumIncrements(nums = [1, 4, 9, 16, 25, 36, 49, 64, 81, 100], target = [8, 27]) == 0

Answer: assert Solution().minimumIncrements(nums = [1, 3, 5, 7, 9, 11, 13, 15, 17, 19], target = [2, 4, 8, 16]) == 1

Answer: assert Solution().minimumIncrements(nums = [11,22,33,44,55,66,77,88,99], target = [11,22,33]) == 0

Answer: assert Solution().minimumIncrements(nums = [2,4,6,8,10,12,14,16,18,20], target = [15,30]) == 10

Answer: assert Solution().minimumIncrements(nums = [1,3,5,7,9,11,13,15,17,19], target = [20,10]) == 1

Answer: assert Solution().minimumIncrements(nums = [7,14,21,28,35], target = [28,35]) == 0

Answer: assert Solution().minimumIncrements(nums = [8,16,24,32,40], target = [16,32,48]) == 8

Answer: assert Solution().minimumIncrements(nums = [7,14,21,28,35,42,49,56,63,70], target = [42,56,70]) == 0

Answer: assert Solution().minimumIncrements(nums = [9, 18, 27, 36, 45, 54, 63, 72, 81, 90], target = [10, 20, 30]) == 2

Answer: assert Solution().minimumIncrements(nums = [13, 26, 39, 52, 65], target = [13, 26, 39, 52, 65]) == 0

Answer: assert Solution().minimumIncrements(nums = [7, 14, 21, 28, 35], target = [14, 28, 42]) == 7

Answer: assert Solution().minimumIncrements(nums = [4, 8, 12, 16], target = [9, 15]) == 4

Answer: assert Solution().minimumIncrements(nums = [3, 6, 9, 12, 15, 18, 21, 24, 27, 30], target = [12, 18, 24]) == 0

Answer: assert Solution().minimumIncrements(nums = [4,8,12,16,20,24,28,32,36,40], target = [7,14,21]) == 1

Answer: assert Solution().minimumIncrements(nums = [3, 6, 9, 12, 15], target = [18, 21]) == 12

Answer: assert Solution().minimumIncrements(nums = [2, 5, 8, 11, 14], target = [11, 22]) == 8

Answer: assert Solution().minimumIncrements(nums = [17, 34, 51, 68], target = [68, 85]) == 34

Answer: assert Solution().minimumIncrements(nums = [12, 24, 36, 48, 60], target = [48, 60]) == 0

Answer: assert Solution().minimumIncrements(nums = [13,26,39,52,65,78,91,104,117,130], target = [13,26,39]) == 0

Answer: assert Solution().minimumIncrements(nums = [2, 3, 5, 7, 11], target = [6, 10]) == 4

Answer: assert Solution().minimumIncrements(nums = [5,10,15,20,25,30], target = [25,35]) == 5

Answer: assert Solution().minimumIncrements(nums = [5,15,25,35,45,55,65,75,85,95], target = [10,20,30]) == 5

Answer: assert Solution().minimumIncrements(nums = [11, 22, 33, 44], target = [33, 44]) == 0

Answer: assert Solution().minimumIncrements(nums = [100, 200, 300, 400, 500, 600, 700, 800, 900, 1000], target = [120, 150, 180, 210]) == 10

Answer: assert Solution().minimumIncrements(nums = [13, 26, 39, 52, 65], target = [52, 65]) == 0

Answer: assert Solution().minimumIncrements(nums = [500, 500, 500, 500, 500, 500, 500, 500, 500, 500], target = [1000, 1500, 2000]) == 2500

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 minimumIncrements(self, nums: list[int], target: list[int]) -> int:
    maxMask = 1 << len(target)
    maskToLcm = {}

    for mask in range(1, maxMask):
      subset = [num for i, num in enumerate(target) if mask >> i & 1]
      maskToLcm[mask] = functools.reduce(math.lcm, subset, 1)

    # dp[mask] := the minimum number of increments to make each number in the
    # subset of target have at least one number that is a multiple in `num`,
    # where `mask` is the bitmask of the subset of target
    dp = [math.inf] * maxMask
    dp[0] = 0

    for num in nums:
      # maskToCost := (mask, cost), where `mask` is the bitmask of the subset
      # of target and `cost` is the minimum number of increments to make each
      # number in the subset of target have at least one number that is a
      # multiple in `num`
      maskToCost = [
          (mask, 0 if (remainder := num % lcm) == 0 else lcm - remainder) for mask,
          lcm in maskToLcm.items()]
      newDp = dp[:]
      for prevMask in range(maxMask):
        if dp[prevMask] == float('inf'):
          continue
        for mask, cost in maskToCost:
          nextMask = prevMask | mask
          newDp[nextMask] = min(newDp[nextMask], dp[prevMask] + cost)
      dp = newDp

    return -1 if dp[-1] == math.inf else dp[-1]