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 source of size n, a string pattern that is a subsequence of source, and a sorted integer array targetIndices that contains distinct numbers in the range [0, n - 1].
We define an operation as removing a character at an index idx from source such that:

idx is an element of targetIndices.
pattern remains a subsequence of source after removing the character.

Performing an operation does not change the indices of the other characters in source. For example, if you remove 'c' from "acb", the character at index 2 would still be 'b'.
Return the maximum number of operations that can be performed.

Example 1:

Input: source = "abbaa", pattern = "aba", targetIndices = [0,1,2]
Output: 1
Explanation:
We can't remove source[0] but we can do either of these two operations:

Remove source[1], so that source becomes "a_baa".
Remove source[2], so that source becomes "ab_aa".

Example 2:

Input: source = "bcda", pattern = "d", targetIndices = [0,3]
Output: 2
Explanation:
We can remove source[0] and source[3] in two operations.

Example 3:

Input: source = "dda", pattern = "dda", targetIndices = [0,1,2]
Output: 0
Explanation:
We can't remove any character from source.

Example 4:

Input: source = "yeyeykyded", pattern = "yeyyd", targetIndices = [0,2,3,4]
Output: 2
Explanation:
We can remove source[2] and source[3] in two operations.

Constraints:

1 <= n == source.length <= 3 * 103
1 <= pattern.length <= n
1 <= targetIndices.length <= n
targetIndices is sorted in ascending order.
The input is generated such that targetIndices contains distinct elements in the range [0, n - 1].
source and pattern consist only of lowercase English letters.
The input is generated such that pattern appears as a subsequence in source.

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

SOLUTION:

Question:
PROBLEM:
You are given a string source of size n, a string pattern that is a subsequence of source, and a sorted integer array targetIndices that contains distinct numbers in the range [0, n - 1].
We define an operation as removing a character at an index idx from source such that:

idx is an element of targetIndices.
pattern remains a subsequence of source after removing the character.

Performing an operation does not change the indices of the other characters in source. For example, if you remove 'c' from "acb", the character at index 2 would still be 'b'.
Return the maximum number of operations that can be performed.

Example 1:

Input: source = "abbaa", pattern = "aba", targetIndices = [0,1,2]
Output: 1
Explanation:
We can't remove source[0] but we can do either of these two operations:

Remove source[1], so that source becomes "a_baa".
Remove source[2], so that source becomes "ab_aa".

Example 2:

Input: source = "bcda", pattern = "d", targetIndices = [0,3]
Output: 2
Explanation:
We can remove source[0] and source[3] in two operations.

Example 3:

Input: source = "dda", pattern = "dda", targetIndices = [0,1,2]
Output: 0
Explanation:
We can't remove any character from source.

Example 4:

Input: source = "yeyeykyded", pattern = "yeyyd", targetIndices = [0,2,3,4]
Output: 2
Explanation:
We can remove source[2] and source[3] in two operations.

Constraints:

1 <= n == source.length <= 3 * 103
1 <= pattern.length <= n
1 <= targetIndices.length <= n
targetIndices is sorted in ascending order.
The input is generated such that targetIndices contains distinct elements in the range [0, n - 1].
source and pattern consist only of lowercase English letters.
The input is generated such that pattern appears as a subsequence in source.

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

SOLUTION:

assert Solution().maxRemovals(source = "abbaa", pattern = "aba", targetIndices = [0,1,2]) == 1

assert Solution().maxRemovals(source = "abracadabra", pattern = "abc", targetIndices = [0,1,5,7,10]) == 3

assert Solution().maxRemovals(source = "bcda", pattern = "d", targetIndices = [0,3]) == 2

assert Solution().maxRemovals(source = "aaaaa", pattern = "aaa", targetIndices = [0,1,2,3,4]) == 2

assert Solution().maxRemovals(source = "dda", pattern = "dda", targetIndices = [0,1,2]) == 0

assert Solution().maxRemovals(source = "abcd", pattern = "ac", targetIndices = [0,1,2,3]) == 2

assert Solution().maxRemovals(source = "zzzz", pattern = "z", targetIndices = [0,1,2,3]) == 3

assert Solution().maxRemovals(source = "aaaaabbbbb", pattern = "ab", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 8

assert Solution().maxRemovals(source = "mississippi", pattern = "issi", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12]) == 7

assert Solution().maxRemovals(source = "abcabcabc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8]) == 6

assert Solution().maxRemovals(source = "abcde", pattern = "ae", targetIndices = [1,2,3]) == 3

assert Solution().maxRemovals(source = "yeyeykyded", pattern = "yeyyd", targetIndices = [0,2,3,4]) == 2

assert Solution().maxRemovals(source = "abracadabra", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10]) == 8

assert Solution().maxRemovals(source = "abcdefghij", pattern = "aceg", targetIndices = [0,2,4,6,8]) == 1

assert Solution().maxRemovals(source = "aaaaabbbbbccccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 12

assert Solution().maxRemovals(source = "abcdefghijk", pattern = "acegik", targetIndices = [1,3,5,7,9]) == 5

assert Solution().maxRemovals(source = "xyzxyzxyz", pattern = "xyz", targetIndices = [1,2,4,5,7,8]) == 4

assert Solution().maxRemovals(source = "aaaaabbbbbcccccdddddeeeeeffffffggggghhhhhh", pattern = "abcdefg", targetIndices = [0,5,10,15,20,25,30,35,40,45]) == 9

assert Solution().maxRemovals(source = "abcabcabcabc", pattern = "abcabc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 6

assert Solution().maxRemovals(source = "abcdabcdabcdabcd", pattern = "abcdabcd", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 8

assert Solution().maxRemovals(source = "xyzxyzxyz", pattern = "xyz", targetIndices = [0,3,6,9,12,15]) == 2

assert Solution().maxRemovals(source = "abcdefghijabcdefghij", pattern = "abcdefghij", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]) == 10

assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "aca", targetIndices = [0,2,4,6,8,10,12,14]) == 6

assert Solution().maxRemovals(source = "abacabacabacaba", pattern = "acaba", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 10

assert Solution().maxRemovals(source = "zzyzxzyzxzyz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 9

assert Solution().maxRemovals(source = "aabbccddeeff", pattern = "abcdef", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 6

assert Solution().maxRemovals(source = "abcabcabcabcabc", pattern = "abcabc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 9

assert Solution().maxRemovals(source = "zzzzzzzzzz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

assert Solution().maxRemovals(source = "abcdabcdabcdabcd", pattern = "abcd", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13]) == 12

assert Solution().maxRemovals(source = "abcdabcdabcdabcd", pattern = "abdc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13]) == 11

assert Solution().maxRemovals(source = "aaaaaabbbbcccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 11

assert Solution().maxRemovals(source = "ababababab", pattern = "aba", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

assert Solution().maxRemovals(source = "zzzzzzzzzz", pattern = "zzzz", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 6

assert Solution().maxRemovals(source = "ababababab", pattern = "aaaa", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 6

assert Solution().maxRemovals(source = "aaaaaaaab", pattern = "a", targetIndices = [0,1,2,3,4,5,6,7]) == 7

assert Solution().maxRemovals(source = "aaaabbbbccccdddd", pattern = "abcd", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) == 12

assert Solution().maxRemovals(source = "xxyyzzxxyyzz", pattern = "xyz", targetIndices = [0,2,4,6,8,10]) == 6

assert Solution().maxRemovals(source = "abracadabra", pattern = "abra", targetIndices = [0,1,2,3,5,6,8,9,10]) == 6

assert Solution().maxRemovals(source = "mississippiissippi", pattern = "issi", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]) == 14

assert Solution().maxRemovals(source = "aaaabbbbcccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 9

assert Solution().maxRemovals(source = "abcdefgabcdefg", pattern = "abcde", targetIndices = [0,1,2,3,4,8,9,10,11,12]) == 6

assert Solution().maxRemovals(source = "abcdefghijklmnopqrstuvwxyz", pattern = "acegikmoqsuwy", targetIndices = [0,2,4,6,8,10,12,14,16,18,20,22,24]) == 0

assert Solution().maxRemovals(source = "qwertyuiopasdfghjklzxcvbnmqwertyuiopasdfghjklzxcvbnm", pattern = "qwerty", targetIndices = [0,1,2,3,4,5,10,11,12,13,14,15,20,21,22,23,24,25,30,31,32,33,34,35]) == 22

assert Solution().maxRemovals(source = "abcdefghijklmnopqrstuvwxyz", pattern = "adgjmpsvy", targetIndices = [1,3,5,7,9,11,13,15,17,19,21,23,25]) == 9

assert Solution().maxRemovals(source = "xyzzxyzzxyzz", pattern = "xzy", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 9

assert Solution().maxRemovals(source = "abcdefghijklmnopqrstuvwxyz", pattern = "acegikmoqsuwy", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]) == 13

assert Solution().maxRemovals(source = "aabbbcccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8]) == 6

assert Solution().maxRemovals(source = "aaaaaaaaaaa", pattern = "aaaa", targetIndices = [1,2,3,5,6,7,9,10]) == 7

assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "abc", targetIndices = [0,2,4,6,8,10,12,14]) == 7

assert Solution().maxRemovals(source = "abcdefghijabcdefghij", pattern = "acegi", targetIndices = [0,2,4,6,8,10,12,14,16,18]) == 5

assert Solution().maxRemovals(source = "mnopqrstuvwxyzabcdefghijkl", pattern = "mnopqrstuvwxyz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]) == 12

assert Solution().maxRemovals(source = "aaaaaa", pattern = "a", targetIndices = [0,1,2,3,4,5]) == 5

assert Solution().maxRemovals(source = "abcdabcdabcd", pattern = "abcd", targetIndices = [0,1,2,3,8,9,10,11]) == 8

assert Solution().maxRemovals(source = "banana", pattern = "baa", targetIndices = [0,1,2,3,4,5]) == 3

assert Solution().maxRemovals(source = "abracadabra", pattern = "abracadabra", targetIndices = [0,1,2,3,4,5,6,7,8,9,10]) == 0

assert Solution().maxRemovals(source = "xyzzazxzyz", pattern = "xyz", targetIndices = [0,2,3,4,5,6,7,8,9]) == 7

assert Solution().maxRemovals(source = "zxyxyxyxyxyxyxyxyxyxyxzyxzyxzyxzyxzyxzyx", pattern = "zyxzyxzyxzyx", targetIndices = [0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46]) == 15

assert Solution().maxRemovals(source = "pppppqqqqqrrrrrssssstttttuuuuuvvvvvwwwwwxxxxxyyyyyzzzzz", pattern = "pqrsuvwxzy", targetIndices = [0,5,10,15,20,25,30,35,40,45,50,55,60,65,70]) == -inf

assert Solution().maxRemovals(source = "zzzzzzzzzzz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10]) == 8

assert Solution().maxRemovals(source = "abracadabra", pattern = "aca", targetIndices = [0,2,5,8,10]) == 5

assert Solution().maxRemovals(source = "abcdabcdeabcdabcde", pattern = "abcde", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]) == 13

assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "ababa", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12]) == 10

assert Solution().maxRemovals(source = "programming", pattern = "pin", targetIndices = [0,2,3,5,7,9,10,11]) == 5

assert Solution().maxRemovals(source = "abcdefghij", pattern = "acegi", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 5

assert Solution().maxRemovals(source = "xyxxyxyxyx", pattern = "xyx", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

assert Solution().maxRemovals(source = "xyxyxyxyxyxyxy", pattern = "xyxy", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 10

assert Solution().maxRemovals(source = "mamamamamamama", pattern = "mam", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 10

assert Solution().maxRemovals(source = "algorithms", pattern = "als", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

assert Solution().maxRemovals(source = "aabbaabbaabbaabb", pattern = "aab", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) == 13

assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 12

assert Solution().maxRemovals(source = "aabbccddeeffgg", pattern = "abcdefg", targetIndices = [0,2,4,6,8,10,12,14]) == 7

Answer: assert Solution().maxRemovals(source = "abbaa", pattern = "aba", targetIndices = [0,1,2]) == 1

Answer: assert Solution().maxRemovals(source = "abracadabra", pattern = "abc", targetIndices = [0,1,5,7,10]) == 3

Answer: assert Solution().maxRemovals(source = "bcda", pattern = "d", targetIndices = [0,3]) == 2

Answer: assert Solution().maxRemovals(source = "aaaaa", pattern = "aaa", targetIndices = [0,1,2,3,4]) == 2

Answer: assert Solution().maxRemovals(source = "dda", pattern = "dda", targetIndices = [0,1,2]) == 0

Answer: assert Solution().maxRemovals(source = "abcd", pattern = "ac", targetIndices = [0,1,2,3]) == 2

Answer: assert Solution().maxRemovals(source = "zzzz", pattern = "z", targetIndices = [0,1,2,3]) == 3

Answer: assert Solution().maxRemovals(source = "aaaaabbbbb", pattern = "ab", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 8

Answer: assert Solution().maxRemovals(source = "mississippi", pattern = "issi", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12]) == 7

Answer: assert Solution().maxRemovals(source = "abcabcabc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8]) == 6

Answer: assert Solution().maxRemovals(source = "abcde", pattern = "ae", targetIndices = [1,2,3]) == 3

Answer: assert Solution().maxRemovals(source = "yeyeykyded", pattern = "yeyyd", targetIndices = [0,2,3,4]) == 2

Answer: assert Solution().maxRemovals(source = "abracadabra", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10]) == 8

Answer: assert Solution().maxRemovals(source = "abcdefghij", pattern = "aceg", targetIndices = [0,2,4,6,8]) == 1

Answer: assert Solution().maxRemovals(source = "ababababababababababababababababababababa", pattern = "abab", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]) == 36

Answer: assert Solution().maxRemovals(source = "aaaaabbbbbccccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 12

Answer: assert Solution().maxRemovals(source = "abcdefghijk", pattern = "acegik", targetIndices = [1,3,5,7,9]) == 5

Answer: assert Solution().maxRemovals(source = "xyzxyzxyz", pattern = "xyz", targetIndices = [1,2,4,5,7,8]) == 4

Answer: assert Solution().maxRemovals(source = "aaaaabbbbbcccccdddddeeeeeffffffggggghhhhhh", pattern = "abcdefg", targetIndices = [0,5,10,15,20,25,30,35,40,45]) == 9

Answer: assert Solution().maxRemovals(source = "abcabcabcabc", pattern = "abcabc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 6

Answer: assert Solution().maxRemovals(source = "abcdabcdabcdabcd", pattern = "abcdabcd", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 8

Answer: assert Solution().maxRemovals(source = "aabbccddeeffgghhiijjkkllmmnnooppqqrrssttuuvvwwxxyyzz", pattern = "abcdefghijklmnopqrstuvwxyz", targetIndices = [0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,98]) == 26

Answer: assert Solution().maxRemovals(source = "xyzxyzxyz", pattern = "xyz", targetIndices = [0,3,6,9,12,15]) == 2

Answer: assert Solution().maxRemovals(source = "abcdefghijabcdefghij", pattern = "abcdefghij", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]) == 10

Answer: assert Solution().maxRemovals(source = "xxyyzzxxyyzzxxyyzzxxyyzzxxyyzzxxyyzz", pattern = "xyz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]) == 33

Answer: assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "aca", targetIndices = [0,2,4,6,8,10,12,14]) == 6

Answer: assert Solution().maxRemovals(source = "abacabacabacaba", pattern = "acaba", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 10

Answer: assert Solution().maxRemovals(source = "zzyzxzyzxzyz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 9

Answer: assert Solution().maxRemovals(source = "helloworldhelloworldhelloworld", pattern = "howrd", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]) == 25

Answer: assert Solution().maxRemovals(source = "aabbccddeeff", pattern = "abcdef", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 6

Answer: assert Solution().maxRemovals(source = "abcabcabcabcabc", pattern = "abcabc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 9

Answer: assert Solution().maxRemovals(source = "ababababababababababababababababababababa", pattern = "aab", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39]) == 37

Answer: assert Solution().maxRemovals(source = "zzzzzzzzzz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

Answer: assert Solution().maxRemovals(source = "abababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababababab", pattern = "aba", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99]) == 100

Answer: assert Solution().maxRemovals(source = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99]) == 99

Answer: assert Solution().maxRemovals(source = "abcdabcdabcdabcd", pattern = "abcd", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13]) == 12

Answer: assert Solution().maxRemovals(source = "abcdabcdabcdabcd", pattern = "abdc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13]) == 11

Answer: assert Solution().maxRemovals(source = "aaaaaabbbbcccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 11

Answer: assert Solution().maxRemovals(source = "ababababab", pattern = "aba", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

Answer: assert Solution().maxRemovals(source = "abcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcdabcd", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99]) == 100

Answer: assert Solution().maxRemovals(source = "zzzzzzzzzz", pattern = "zzzz", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 6

Answer: assert Solution().maxRemovals(source = "abcabcabcabcabcabcabcabcabcabc", pattern = "abcabc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]) == 24

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

Answer: assert Solution().maxRemovals(source = "pneumonoultramicroscopicsilicovolcanoconiosis", pattern = "pum", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]) == 42

Answer: assert Solution().maxRemovals(source = "ababababab", pattern = "aaaa", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 6

Answer: assert Solution().maxRemovals(source = "aaaaaaaab", pattern = "a", targetIndices = [0,1,2,3,4,5,6,7]) == 7

Answer: assert Solution().maxRemovals(source = "aaaabbbbccccdddd", pattern = "abcd", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) == 12

Answer: assert Solution().maxRemovals(source = "xxyyzzxxyyzz", pattern = "xyz", targetIndices = [0,2,4,6,8,10]) == 6

Answer: assert Solution().maxRemovals(source = "abracadabra", pattern = "abra", targetIndices = [0,1,2,3,5,6,8,9,10]) == 6

Answer: assert Solution().maxRemovals(source = "mississippiissippi", pattern = "issi", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]) == 14

Answer: assert Solution().maxRemovals(source = "aaaabbbbcccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 9

Answer: assert Solution().maxRemovals(source = "abcdefgabcdefg", pattern = "abcde", targetIndices = [0,1,2,3,4,8,9,10,11,12]) == 6

Answer: assert Solution().maxRemovals(source = "abcdefghijklmnopqrstuvwxyz", pattern = "acegikmoqsuwy", targetIndices = [0,2,4,6,8,10,12,14,16,18,20,22,24]) == 0

Answer: assert Solution().maxRemovals(source = "qwertyuiopasdfghjklzxcvbnmqwertyuiopasdfghjklzxcvbnm", pattern = "qwerty", targetIndices = [0,1,2,3,4,5,10,11,12,13,14,15,20,21,22,23,24,25,30,31,32,33,34,35]) == 22

Answer: assert Solution().maxRemovals(source = "abcdefghijklmnopqrstuvwxyz", pattern = "adgjmpsvy", targetIndices = [1,3,5,7,9,11,13,15,17,19,21,23,25]) == 9

Answer: assert Solution().maxRemovals(source = "aabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabbaabb", pattern = "aabbaabbaabb", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75]) == 56

Answer: assert Solution().maxRemovals(source = "xyzzxyzzxyzz", pattern = "xzy", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 9

Answer: assert Solution().maxRemovals(source = "abcdefghijklmnopqrstuvwxyz", pattern = "acegikmoqsuwy", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]) == 13

Answer: assert Solution().maxRemovals(source = "aabbbcccc", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8]) == 6

Answer: assert Solution().maxRemovals(source = "aaaaaaaaaaa", pattern = "aaaa", targetIndices = [1,2,3,5,6,7,9,10]) == 7

Answer: assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "abc", targetIndices = [0,2,4,6,8,10,12,14]) == 7

Answer: assert Solution().maxRemovals(source = "abcdefghijabcdefghij", pattern = "acegi", targetIndices = [0,2,4,6,8,10,12,14,16,18]) == 5

Answer: assert Solution().maxRemovals(source = "mnopqrstuvwxyzabcdefghijkl", pattern = "mnopqrstuvwxyz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]) == 12

Answer: assert Solution().maxRemovals(source = "aaaaaa", pattern = "a", targetIndices = [0,1,2,3,4,5]) == 5

Answer: assert Solution().maxRemovals(source = "abcdabcdabcd", pattern = "abcd", targetIndices = [0,1,2,3,8,9,10,11]) == 8

Answer: assert Solution().maxRemovals(source = "banana", pattern = "baa", targetIndices = [0,1,2,3,4,5]) == 3

Answer: assert Solution().maxRemovals(source = "abracadabra", pattern = "abracadabra", targetIndices = [0,1,2,3,4,5,6,7,8,9,10]) == 0

Answer: assert Solution().maxRemovals(source = "xyzzazxzyz", pattern = "xyz", targetIndices = [0,2,3,4,5,6,7,8,9]) == 7

Answer: assert Solution().maxRemovals(source = "zxyxyxyxyxyxyxyxyxyxyxzyxzyxzyxzyxzyxzyx", pattern = "zyxzyxzyxzyx", targetIndices = [0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46]) == 15

Answer: assert Solution().maxRemovals(source = "pppppqqqqqrrrrrssssstttttuuuuuvvvvvwwwwwxxxxxyyyyyzzzzz", pattern = "pqrsuvwxzy", targetIndices = [0,5,10,15,20,25,30,35,40,45,50,55,60,65,70]) == -inf

Answer: assert Solution().maxRemovals(source = "zzzzzzzzzzz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10]) == 8

Answer: assert Solution().maxRemovals(source = "abracadabra", pattern = "aca", targetIndices = [0,2,5,8,10]) == 5

Answer: assert Solution().maxRemovals(source = "abcdabcdeabcdabcde", pattern = "abcde", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19]) == 13

Answer: assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "ababa", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12]) == 10

Answer: assert Solution().maxRemovals(source = "programming", pattern = "pin", targetIndices = [0,2,3,5,7,9,10,11]) == 5

Answer: assert Solution().maxRemovals(source = "abcdefghij", pattern = "acegi", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 5

Answer: assert Solution().maxRemovals(source = "zzzzzzzzzzzzzzzzzzzzzzzzzz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]) == 23

Answer: assert Solution().maxRemovals(source = "xyxxyxyxyx", pattern = "xyx", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

Answer: assert Solution().maxRemovals(source = "xyxyxyxyxyxyxy", pattern = "xyxy", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 10

Answer: assert Solution().maxRemovals(source = "mamamamamamama", pattern = "mam", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11]) == 10

Answer: assert Solution().maxRemovals(source = "xyzyzyzyzyzyzyzyzyzyzyzyzyzyzyzyzyzyzyzyzyzyz", pattern = "zyzyz", targetIndices = [0,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60]) == 20

Answer: assert Solution().maxRemovals(source = "algorithms", pattern = "als", targetIndices = [0,1,2,3,4,5,6,7,8,9]) == 7

Answer: assert Solution().maxRemovals(source = "aabbaabbaabbaabb", pattern = "aab", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]) == 13

Answer: assert Solution().maxRemovals(source = "abacabadabacaba", pattern = "abc", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14]) == 12

Answer: assert Solution().maxRemovals(source = "zzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzzz", pattern = "zzz", targetIndices = [0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]) == 50

Answer: assert Solution().maxRemovals(source = "aabbccddeeffgg", pattern = "abcdefg", targetIndices = [0,2,4,6,8,10,12,14]) == 7

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 maxRemovals(self, source: str, pattern: str, targetIndices: List[int]) -> int:
        m, n = len(source), len(pattern)
        f = [[-inf] * (n + 1) for _ in range(m + 1)]
        f[0][0] = 0
        s = set(targetIndices)
        for i, c in enumerate(source, 1):
            for j in range(n + 1):
                f[i][j] = f[i - 1][j] + int((i - 1) in s)
                if j and c == pattern[j - 1]:
                    f[i][j] = max(f[i][j], f[i - 1][j - 1])
        return f[m][n]

class Solution:
    def maxRemovals(self, source: str, pattern: str, targetIndices: List[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 three integers n, x, and y.
An event is being held for n performers. When a performer arrives, they are assigned to one of the x stages. All performers assigned to the same stage will perform together as a band, though some stages might remain empty.
After all performances are completed, the jury will award each band a score in the range [1, y].
Return the total number of possible ways the event can take place.
Since the answer may be very large, return it modulo 109 + 7.
Note that two events are considered to have been held differently if either of the following conditions is satisfied:

Any performer is assigned a different stage.
Any band is awarded a different score.

Example 1:

Input: n = 1, x = 2, y = 3
Output: 6
Explanation:

There are 2 ways to assign a stage to the performer.
The jury can award a score of either 1, 2, or 3 to the only band.

Example 2:

Input: n = 5, x = 2, y = 1
Output: 32
Explanation:

Each performer will be assigned either stage 1 or stage 2.
All bands will be awarded a score of 1.

Example 3:

Input: n = 3, x = 3, y = 4
Output: 684

Constraints:

1 <= n, x, y <= 1000

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

SOLUTION:

Question:
PROBLEM:
You are given three integers n, x, and y.
An event is being held for n performers. When a performer arrives, they are assigned to one of the x stages. All performers assigned to the same stage will perform together as a band, though some stages might remain empty.
After all performances are completed, the jury will award each band a score in the range [1, y].
Return the total number of possible ways the event can take place.
Since the answer may be very large, return it modulo 109 + 7.
Note that two events are considered to have been held differently if either of the following conditions is satisfied:

Any performer is assigned a different stage.
Any band is awarded a different score.

Example 1:

Input: n = 1, x = 2, y = 3
Output: 6
Explanation:

There are 2 ways to assign a stage to the performer.
The jury can award a score of either 1, 2, or 3 to the only band.

Example 2:

Input: n = 5, x = 2, y = 1
Output: 32
Explanation:

Each performer will be assigned either stage 1 or stage 2.
All bands will be awarded a score of 1.

Example 3:

Input: n = 3, x = 3, y = 4
Output: 684

Constraints:

1 <= n, x, y <= 1000

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

SOLUTION:

assert Solution().numberOfWays(n = 6, x = 6, y = 6) == 150265836

assert Solution().numberOfWays(n = 4, x = 2, y = 4) == 232

assert Solution().numberOfWays(n = 500, x = 5, y = 20) == 955685646

assert Solution().numberOfWays(n = 100, x = 10, y = 10) == 757713628

assert Solution().numberOfWays(n = 1, x = 2, y = 3) == 6

assert Solution().numberOfWays(n = 5, x = 2, y = 1) == 32

assert Solution().numberOfWays(n = 1, x = 1, y = 1) == 1

assert Solution().numberOfWays(n = 7, x = 4, y = 3) == 882264

assert Solution().numberOfWays(n = 10, x = 10, y = 1) == 999999937

assert Solution().numberOfWays(n = 7, x = 1, y = 10) == 10

assert Solution().numberOfWays(n = 2, x = 3, y = 2) == 30

assert Solution().numberOfWays(n = 100, x = 10, y = 100) == 273493465

assert Solution().numberOfWays(n = 500, x = 500, y = 1) == 742761597

assert Solution().numberOfWays(n = 10, x = 5, y = 5) == 574980399

assert Solution().numberOfWays(n = 10, x = 5, y = 10) == 786818193

assert Solution().numberOfWays(n = 2, x = 2, y = 2) == 12

assert Solution().numberOfWays(n = 3, x = 3, y = 4) == 684

assert Solution().numberOfWays(n = 2, x = 4, y = 3) == 120

assert Solution().numberOfWays(n = 500, x = 20, y = 30) == 969962044

assert Solution().numberOfWays(n = 7, x = 3, y = 2) == 15966

assert Solution().numberOfWays(n = 2, x = 2, y = 1000) == 2002000

assert Solution().numberOfWays(n = 500, x = 500, y = 10) == 945662039

assert Solution().numberOfWays(n = 800, x = 25, y = 25) == 241877807

assert Solution().numberOfWays(n = 750, x = 25, y = 35) == 837224424

assert Solution().numberOfWays(n = 100, x = 50, y = 50) == 54657599

assert Solution().numberOfWays(n = 250, x = 5, y = 200) == 919377263

assert Solution().numberOfWays(n = 1000, x = 10, y = 10) == 187897528

assert Solution().numberOfWays(n = 999, x = 999, y = 1) == 760074701

assert Solution().numberOfWays(n = 400, x = 400, y = 2) == 877938741

assert Solution().numberOfWays(n = 100, x = 100, y = 1) == 424090053

assert Solution().numberOfWays(n = 750, x = 30, y = 40) == 725279144

assert Solution().numberOfWays(n = 1000, x = 1, y = 1000) == 1000

assert Solution().numberOfWays(n = 600, x = 50, y = 100) == 201548400

assert Solution().numberOfWays(n = 20, x = 10, y = 10) == 250624124

assert Solution().numberOfWays(n = 800, x = 2, y = 3) == 105983888

assert Solution().numberOfWays(n = 600, x = 3, y = 500) == 507224996

assert Solution().numberOfWays(n = 250, x = 250, y = 250) == 37703966

assert Solution().numberOfWays(n = 200, x = 1000, y = 1) == 500816181

assert Solution().numberOfWays(n = 200, x = 100, y = 1) == 794576212

assert Solution().numberOfWays(n = 10, x = 1, y = 1000) == 1000

assert Solution().numberOfWays(n = 400, x = 40, y = 40) == 130346951

assert Solution().numberOfWays(n = 500, x = 3, y = 5) == 954723270

assert Solution().numberOfWays(n = 1000, x = 50, y = 50) == 429860379

assert Solution().numberOfWays(n = 300, x = 2, y = 100) == 507547660

assert Solution().numberOfWays(n = 1, x = 1000, y = 1000) == 1000000

assert Solution().numberOfWays(n = 10, x = 50, y = 100) == 512711582

assert Solution().numberOfWays(n = 10, x = 100, y = 50) == 634873075

assert Solution().numberOfWays(n = 50, x = 5, y = 10) == 182572441

assert Solution().numberOfWays(n = 50, x = 10, y = 20) == 139021947

assert Solution().numberOfWays(n = 50, x = 10, y = 5) == 722025534

assert Solution().numberOfWays(n = 800, x = 800, y = 2) == 34432269

assert Solution().numberOfWays(n = 999, x = 1, y = 999) == 999

assert Solution().numberOfWays(n = 500, x = 10, y = 100) == 266610171

assert Solution().numberOfWays(n = 700, x = 7, y = 7) == 275200648

assert Solution().numberOfWays(n = 500, x = 50, y = 50) == 25088782

assert Solution().numberOfWays(n = 900, x = 50, y = 50) == 414449887

assert Solution().numberOfWays(n = 2, x = 1000, y = 2) == 3998000

assert Solution().numberOfWays(n = 300, x = 1, y = 1000) == 1000

assert Solution().numberOfWays(n = 1000, x = 2, y = 2) == 753692822

assert Solution().numberOfWays(n = 500, x = 1, y = 1000) == 1000

Answer: assert Solution().numberOfWays(n = 500, x = 250, y = 250) == 604049962

Answer: assert Solution().numberOfWays(n = 6, x = 6, y = 6) == 150265836

Answer: assert Solution().numberOfWays(n = 4, x = 2, y = 4) == 232

Answer: assert Solution().numberOfWays(n = 500, x = 5, y = 20) == 955685646

Answer: assert Solution().numberOfWays(n = 100, x = 10, y = 10) == 757713628

Answer: assert Solution().numberOfWays(n = 1, x = 2, y = 3) == 6

Answer: assert Solution().numberOfWays(n = 1000, x = 1000, y = 1000) == 295964505

Answer: assert Solution().numberOfWays(n = 5, x = 2, y = 1) == 32

Answer: assert Solution().numberOfWays(n = 1, x = 1, y = 1) == 1

Answer: assert Solution().numberOfWays(n = 7, x = 4, y = 3) == 882264

Answer: assert Solution().numberOfWays(n = 10, x = 10, y = 1) == 999999937

Answer: assert Solution().numberOfWays(n = 7, x = 1, y = 10) == 10

Answer: assert Solution().numberOfWays(n = 2, x = 3, y = 2) == 30

Answer: assert Solution().numberOfWays(n = 100, x = 10, y = 100) == 273493465

Answer: assert Solution().numberOfWays(n = 500, x = 500, y = 1) == 742761597

Answer: assert Solution().numberOfWays(n = 10, x = 5, y = 5) == 574980399

Answer: assert Solution().numberOfWays(n = 10, x = 5, y = 10) == 786818193

Answer: assert Solution().numberOfWays(n = 2, x = 2, y = 2) == 12

Answer: assert Solution().numberOfWays(n = 3, x = 3, y = 4) == 684

Answer: assert Solution().numberOfWays(n = 2, x = 4, y = 3) == 120

Answer: assert Solution().numberOfWays(n = 500, x = 20, y = 30) == 969962044

Answer: assert Solution().numberOfWays(n = 7, x = 3, y = 2) == 15966

Answer: assert Solution().numberOfWays(n = 1000, x = 100, y = 100) == 31082313

Answer: assert Solution().numberOfWays(n = 999, x = 1000, y = 999) == 529470001

Answer: assert Solution().numberOfWays(n = 300, x = 100, y = 200) == 920174716

Answer: assert Solution().numberOfWays(n = 500, x = 700, y = 300) == 848923924

Answer: assert Solution().numberOfWays(n = 2, x = 2, y = 1000) == 2002000

Answer: assert Solution().numberOfWays(n = 100, x = 100, y = 100) == 169878723

Answer: assert Solution().numberOfWays(n = 1000, x = 10, y = 100) == 639122056

Answer: assert Solution().numberOfWays(n = 500, x = 500, y = 10) == 945662039

Answer: assert Solution().numberOfWays(n = 600, x = 300, y = 300) == 952131693

Answer: assert Solution().numberOfWays(n = 800, x = 25, y = 25) == 241877807

Answer: assert Solution().numberOfWays(n = 750, x = 25, y = 35) == 837224424

Answer: assert Solution().numberOfWays(n = 100, x = 50, y = 50) == 54657599

Answer: assert Solution().numberOfWays(n = 500, x = 500, y = 500) == 295516381

Answer: assert Solution().numberOfWays(n = 250, x = 5, y = 200) == 919377263

Answer: assert Solution().numberOfWays(n = 999, x = 999, y = 999) == 490429319

Answer: assert Solution().numberOfWays(n = 999, x = 1000, y = 1000) == 78742301

Answer: assert Solution().numberOfWays(n = 500, x = 1000, y = 1000) == 736888411

Answer: assert Solution().numberOfWays(n = 1000, x = 10, y = 10) == 187897528

Answer: assert Solution().numberOfWays(n = 999, x = 999, y = 1) == 760074701

Answer: assert Solution().numberOfWays(n = 400, x = 400, y = 2) == 877938741

Answer: assert Solution().numberOfWays(n = 100, x = 100, y = 1) == 424090053

Answer: assert Solution().numberOfWays(n = 750, x = 30, y = 40) == 725279144

Answer: assert Solution().numberOfWays(n = 300, x = 200, y = 150) == 446039088

Answer: assert Solution().numberOfWays(n = 1000, x = 1, y = 1000) == 1000

Answer: assert Solution().numberOfWays(n = 600, x = 50, y = 100) == 201548400

Answer: assert Solution().numberOfWays(n = 750, x = 500, y = 250) == 201743686

Answer: assert Solution().numberOfWays(n = 20, x = 10, y = 10) == 250624124

Answer: assert Solution().numberOfWays(n = 750, x = 250, y = 350) == 298617350

Answer: assert Solution().numberOfWays(n = 800, x = 2, y = 3) == 105983888

Answer: assert Solution().numberOfWays(n = 450, x = 150, y = 200) == 836208120

Answer: assert Solution().numberOfWays(n = 999, x = 999, y = 1000) == 759466025

Answer: assert Solution().numberOfWays(n = 1000, x = 500, y = 500) == 754687589

Answer: assert Solution().numberOfWays(n = 600, x = 3, y = 500) == 507224996

Answer: assert Solution().numberOfWays(n = 250, x = 250, y = 250) == 37703966

Answer: assert Solution().numberOfWays(n = 200, x = 1000, y = 1) == 500816181

Answer: assert Solution().numberOfWays(n = 200, x = 100, y = 1) == 794576212

Answer: assert Solution().numberOfWays(n = 10, x = 1, y = 1000) == 1000

Answer: assert Solution().numberOfWays(n = 400, x = 40, y = 40) == 130346951

Answer: assert Solution().numberOfWays(n = 500, x = 3, y = 5) == 954723270

Answer: assert Solution().numberOfWays(n = 1000, x = 50, y = 50) == 429860379

Answer: assert Solution().numberOfWays(n = 300, x = 2, y = 100) == 507547660

Answer: assert Solution().numberOfWays(n = 600, x = 300, y = 400) == 520792267

Answer: assert Solution().numberOfWays(n = 1, x = 1000, y = 1000) == 1000000

Answer: assert Solution().numberOfWays(n = 750, x = 250, y = 500) == 534179249

Answer: assert Solution().numberOfWays(n = 10, x = 50, y = 100) == 512711582

Answer: assert Solution().numberOfWays(n = 10, x = 100, y = 50) == 634873075

Answer: assert Solution().numberOfWays(n = 50, x = 5, y = 10) == 182572441

Answer: assert Solution().numberOfWays(n = 750, x = 200, y = 300) == 719876888

Answer: assert Solution().numberOfWays(n = 550, x = 100, y = 100) == 128555335

Answer: assert Solution().numberOfWays(n = 50, x = 10, y = 20) == 139021947

Answer: assert Solution().numberOfWays(n = 1000, x = 200, y = 200) == 985949432

Answer: assert Solution().numberOfWays(n = 800, x = 200, y = 200) == 116750119

Answer: assert Solution().numberOfWays(n = 50, x = 10, y = 5) == 722025534

Answer: assert Solution().numberOfWays(n = 800, x = 800, y = 2) == 34432269

Answer: assert Solution().numberOfWays(n = 999, x = 1, y = 999) == 999

Answer: assert Solution().numberOfWays(n = 500, x = 10, y = 100) == 266610171

Answer: assert Solution().numberOfWays(n = 1000, x = 1000, y = 1) == 524700271

Answer: assert Solution().numberOfWays(n = 800, x = 250, y = 250) == 375598308

Answer: assert Solution().numberOfWays(n = 200, x = 300, y = 100) == 159926237

Answer: assert Solution().numberOfWays(n = 700, x = 7, y = 7) == 275200648

Answer: assert Solution().numberOfWays(n = 999, x = 500, y = 500) == 547290861

Answer: assert Solution().numberOfWays(n = 500, x = 50, y = 50) == 25088782

Answer: assert Solution().numberOfWays(n = 900, x = 50, y = 50) == 414449887

Answer: assert Solution().numberOfWays(n = 500, x = 300, y = 700) == 800306839

Answer: assert Solution().numberOfWays(n = 2, x = 1000, y = 2) == 3998000

Answer: assert Solution().numberOfWays(n = 300, x = 1, y = 1000) == 1000

Answer: assert Solution().numberOfWays(n = 600, x = 100, y = 100) == 172044181

Answer: assert Solution().numberOfWays(n = 1000, x = 2, y = 2) == 753692822

Answer: assert Solution().numberOfWays(n = 500, x = 1, y = 1000) == 1000

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 numberOfWays(self, n: int, x: int, y: int) -> int:
        mod = 10**9 + 7
        f = [[0] * (x + 1) for _ in range(n + 1)]
        f[0][0] = 1
        for i in range(1, n + 1):
            for j in range(1, x + 1):
                f[i][j] = (f[i - 1][j] * j + f[i - 1][j - 1] * (x - (j - 1))) % mod
        ans, p = 0, 1
        for j in range(1, x + 1):
            p = p * y % mod
            ans = (ans + f[n][j] * p) % mod
        return ans


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

PROBLEM:
Alice and Bob are playing a fantasy battle game consisting of n rounds where they summon one of three magical creatures each round: a Fire Dragon, a Water Serpent, or an Earth Golem. In each round, players simultaneously summon their creature and are awarded points as follows:

If one player summons a Fire Dragon and the other summons an Earth Golem, the player who summoned the Fire Dragon is awarded a point.
If one player summons a Water Serpent and the other summons a Fire Dragon, the player who summoned the Water Serpent is awarded a point.
If one player summons an Earth Golem and the other summons a Water Serpent, the player who summoned the Earth Golem is awarded a point.
If both players summon the same creature, no player is awarded a point.

You are given a string s consisting of n characters 'F', 'W', and 'E', representing the sequence of creatures Alice will summon in each round:

If s[i] == 'F', Alice summons a Fire Dragon.
If s[i] == 'W', Alice summons a Water Serpent.
If s[i] == 'E', Alice summons an Earth Golem.

Bob’s sequence of moves is unknown, but it is guaranteed that Bob will never summon the same creature in two consecutive rounds. Bob beats Alice if the total number of points awarded to Bob after n rounds is strictly greater than the points awarded to Alice.
Return the number of distinct sequences Bob can use to beat Alice.
Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: s = "FFF"
Output: 3
Explanation:
Bob can beat Alice by making one of the following sequences of moves: "WFW", "FWF", or "WEW". Note that other winning sequences like "WWE" or "EWW" are invalid since Bob cannot make the same move twice in a row.

Example 2:

Input: s = "FWEFW"
Output: 18
Explanation:
Bob can beat Alice by making one of the following sequences of moves: "FWFWF", "FWFWE", "FWEFE", "FWEWE", "FEFWF", "FEFWE", "FEFEW", "FEWFE", "WFEFE", "WFEWE", "WEFWF", "WEFWE", "WEFEF", "WEFEW", "WEWFW", "WEWFE", "EWFWE", or "EWEWE".

Constraints:

1 <= s.length <= 1000
s[i] is one of 'F', 'W', or 'E'.

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

SOLUTION:

Question:
PROBLEM:
Alice and Bob are playing a fantasy battle game consisting of n rounds where they summon one of three magical creatures each round: a Fire Dragon, a Water Serpent, or an Earth Golem. In each round, players simultaneously summon their creature and are awarded points as follows:

If one player summons a Fire Dragon and the other summons an Earth Golem, the player who summoned the Fire Dragon is awarded a point.
If one player summons a Water Serpent and the other summons a Fire Dragon, the player who summoned the Water Serpent is awarded a point.
If one player summons an Earth Golem and the other summons a Water Serpent, the player who summoned the Earth Golem is awarded a point.
If both players summon the same creature, no player is awarded a point.

You are given a string s consisting of n characters 'F', 'W', and 'E', representing the sequence of creatures Alice will summon in each round:

If s[i] == 'F', Alice summons a Fire Dragon.
If s[i] == 'W', Alice summons a Water Serpent.
If s[i] == 'E', Alice summons an Earth Golem.

Bob’s sequence of moves is unknown, but it is guaranteed that Bob will never summon the same creature in two consecutive rounds. Bob beats Alice if the total number of points awarded to Bob after n rounds is strictly greater than the points awarded to Alice.
Return the number of distinct sequences Bob can use to beat Alice.
Since the answer may be very large, return it modulo 109 + 7.

Example 1:

Input: s = "FFF"
Output: 3
Explanation:
Bob can beat Alice by making one of the following sequences of moves: "WFW", "FWF", or "WEW". Note that other winning sequences like "WWE" or "EWW" are invalid since Bob cannot make the same move twice in a row.

Example 2:

Input: s = "FWEFW"
Output: 18
Explanation:
Bob can beat Alice by making one of the following sequences of moves: "FWFWF", "FWFWE", "FWEFE", "FWEWE", "FEFWF", "FEFWE", "FEFEW", "FEWFE", "WFEFE", "WFEWE", "WEFWF", "WEFWE", "WEFEF", "WEFEW", "WEWFW", "WEWFE", "EWFWE", or "EWEWE".

Constraints:

1 <= s.length <= 1000
s[i] is one of 'F', 'W', or 'E'.

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

SOLUTION:

assert Solution().countWinningSequences(s = "WFWFWFWFWFWFWFWFWFWF") == 730500

assert Solution().countWinningSequences(s = "WFW") == 6

assert Solution().countWinningSequences(s = "FWEFWEFW") == 157

assert Solution().countWinningSequences(s = "FWFWFWFWFWFWFWFWFWFW") == 730500

assert Solution().countWinningSequences(s = "EWE") == 6

assert Solution().countWinningSequences(s = "EEEEEEEEEEEE") == 2364

assert Solution().countWinningSequences(s = "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFWE") == 697784136

assert Solution().countWinningSequences(s = "FWEFWFWFEF") == 671

assert Solution().countWinningSequences(s = "EWFEWFWEWFWEWFWEWFWE") == 723067

assert Solution().countWinningSequences(s = "FFFFFFFFFFFF") == 2364

assert Solution().countWinningSequences(s = "WWE") == 4

assert Solution().countWinningSequences(s = "FWFWFW") == 40

assert Solution().countWinningSequences(s = "FWFEWFWEWF") == 670

assert Solution().countWinningSequences(s = "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF") == 956577544

assert Solution().countWinningSequences(s = "FFF") == 3

assert Solution().countWinningSequences(s = "WFWFWFW") == 86

assert Solution().countWinningSequences(s = "FWEFW") == 18

assert Solution().countWinningSequences(s = "F") == 1

assert Solution().countWinningSequences(s = "EWW") == 4

assert Solution().countWinningSequences(s = "FWFWFWFW") == 168

assert Solution().countWinningSequences(s = "EFEFEF") == 40

assert Solution().countWinningSequences(s = "FFFFFFFFFFFFFFFFFF") == 158972

assert Solution().countWinningSequences(s = "FWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFW") == 380726528

assert Solution().countWinningSequences(s = "EEWEEWEEWE") == 678

assert Solution().countWinningSequences(s = "FWFWFWFWFW") == 688

assert Solution().countWinningSequences(s = "EEEE") == 8

assert Solution().countWinningSequences(s = "FWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWE") == 116417994

Answer: assert Solution().countWinningSequences(s = "WFWFWFWFWFWFWFWFWFWF") == 730500

Answer: assert Solution().countWinningSequences(s = "WFW") == 6

Answer: assert Solution().countWinningSequences(s = "FWEFWEFW") == 157

Answer: assert Solution().countWinningSequences(s = "FWFWFWFWFWFWFWFWFWFW") == 730500

Answer: assert Solution().countWinningSequences(s = "EWE") == 6

Answer: assert Solution().countWinningSequences(s = "WFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF") == 377759202

Answer: assert Solution().countWinningSequences(s = "EEEEEEEEEEEE") == 2364

Answer: assert Solution().countWinningSequences(s = "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFWE") == 697784136

Answer: assert Solution().countWinningSequences(s = "FWEFWFWFEF") == 671

Answer: assert Solution().countWinningSequences(s = "EFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF") == 942021245

Answer: assert Solution().countWinningSequences(s = "EWFEWFWEWFWEWFWEWFWE") == 723067

Answer: assert Solution().countWinningSequences(s = "FFFFFFFFFFFF") == 2364

Answer: assert Solution().countWinningSequences(s = "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF") == 486882611

Answer: assert Solution().countWinningSequences(s = "WWE") == 4

Answer: assert Solution().countWinningSequences(s = "FWFWFW") == 40

Answer: assert Solution().countWinningSequences(s = "FWFEWFWEWF") == 670

Answer: assert Solution().countWinningSequences(s = "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF") == 956577544

Answer: assert Solution().countWinningSequences(s = "FFF") == 3

Answer: assert Solution().countWinningSequences(s = "WFWFWFW") == 86

Answer: assert Solution().countWinningSequences(s = "FWEFW") == 18

Answer: assert Solution().countWinningSequences(s = "F") == 1

Answer: assert Solution().countWinningSequences(s = "EWW") == 4

Answer: assert Solution().countWinningSequences(s = "FWFWFWFW") == 168

Answer: assert Solution().countWinningSequences(s = "EFEFEF") == 40

Answer: assert Solution().countWinningSequences(s = "FFFFFFFFFFFFFFFFFF") == 158972

Answer: assert Solution().countWinningSequences(s = "FWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFW") == 380726528

Answer: assert Solution().countWinningSequences(s = "EEWEEWEEWE") == 678

Answer: assert Solution().countWinningSequences(s = "FWFWFWFWFW") == 688

Answer: assert Solution().countWinningSequences(s = "EEEE") == 8

Answer: assert Solution().countWinningSequences(s = "FWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWFWE") == 116417994

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 countWinningSequences(self, s: str) -> int:
        def calc(x: int, y: int) -> int:
            if x == y:
                return 0
            if x < y:
                return 1 if x == 0 and y == 2 else -1
            return -1 if x == 2 and y == 0 else 1

        @cache
        def dfs(i: int, j: int, k: int) -> int:
            if len(s) - i <= j:
                return 0
            if i >= len(s):
                return int(j < 0)
            res = 0
            for l in range(3):
                if l == k:
                    continue
                res = (res + dfs(i + 1, j + calc(d[s[i]], l), l)) % mod
            return res

        mod = 10**9 + 7
        d = {"F": 0, "W": 1, "E": 2}
        ans = dfs(0, 0, -1)
        dfs.cache_clear()
        return ans


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

PROBLEM:
You are given a 2D array intervals, where intervals[i] = [starti, endi] represents the start and the end of interval i. You are also given an integer k.
You must add exactly one new interval [startnew, endnew] to the array such that:

The length of the new interval, endnew - startnew, is at most k.
After adding, the number of connected groups in intervals is minimized.

A connected group of intervals is a maximal collection of intervals that, when considered together, cover a continuous range from the smallest point to the largest point with no gaps between them. Here are some examples:

A group of intervals [[1, 2], [2, 5], [3, 3]] is connected because together they cover the range from 1 to 5 without any gaps.
However, a group of intervals [[1, 2], [3, 4]] is not connected because the segment (2, 3) is not covered.

Return the minimum number of connected groups after adding exactly one new interval to the array.

Example 1:

Input: intervals = [[1,3],[5,6],[8,10]], k = 3
Output: 2
Explanation:
After adding the interval [3, 5], we have two connected groups: [[1, 3], [3, 5], [5, 6]] and [[8, 10]].

Example 2:

Input: intervals = [[5,10],[1,1],[3,3]], k = 1
Output: 3
Explanation:
After adding the interval [1, 1], we have three connected groups: [[1, 1], [1, 1]], [[3, 3]], and [[5, 10]].

Constraints:

1 <= intervals.length <= 105
intervals[i] == [starti, endi]
1 <= starti <= endi <= 109
1 <= k <= 109

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

SOLUTION:

Question:
PROBLEM:
You are given a 2D array intervals, where intervals[i] = [starti, endi] represents the start and the end of interval i. You are also given an integer k.
You must add exactly one new interval [startnew, endnew] to the array such that:

The length of the new interval, endnew - startnew, is at most k.
After adding, the number of connected groups in intervals is minimized.

A connected group of intervals is a maximal collection of intervals that, when considered together, cover a continuous range from the smallest point to the largest point with no gaps between them. Here are some examples:

A group of intervals [[1, 2], [2, 5], [3, 3]] is connected because together they cover the range from 1 to 5 without any gaps.
However, a group of intervals [[1, 2], [3, 4]] is not connected because the segment (2, 3) is not covered.

Return the minimum number of connected groups after adding exactly one new interval to the array.

Example 1:

Input: intervals = [[1,3],[5,6],[8,10]], k = 3
Output: 2
Explanation:
After adding the interval [3, 5], we have two connected groups: [[1, 3], [3, 5], [5, 6]] and [[8, 10]].

Example 2:

Input: intervals = [[5,10],[1,1],[3,3]], k = 1
Output: 3
Explanation:
After adding the interval [1, 1], we have three connected groups: [[1, 1], [1, 1]], [[3, 3]], and [[5, 10]].

Constraints:

1 <= intervals.length <= 105
intervals[i] == [starti, endi]
1 <= starti <= endi <= 109
1 <= k <= 109

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

SOLUTION:

assert Solution().minConnectedGroups(intervals = [[1,2],[2,3],[3,4]], k = 0) == 1

assert Solution().minConnectedGroups(intervals = [[1,10]], k = 5) == 1

assert Solution().minConnectedGroups(intervals = [[1,2],[4,8],[10,15]], k = 2) == 2

assert Solution().minConnectedGroups(intervals = [[1,2],[4,5],[7,8]], k = 2) == 2

assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20]], k = 4) == 3

assert Solution().minConnectedGroups(intervals = [[1,2],[3,4],[5,6]], k = 1) == 2

assert Solution().minConnectedGroups(intervals = [[5,10],[1,1],[3,3]], k = 1) == 3

assert Solution().minConnectedGroups(intervals = [[1,10],[10,20],[20,30]], k = 5) == 1

assert Solution().minConnectedGroups(intervals = [[1,100]], k = 50) == 1

assert Solution().minConnectedGroups(intervals = [[2,4],[6,8],[10,12]], k = 5) == 2

assert Solution().minConnectedGroups(intervals = [[1,3],[5,6],[8,10]], k = 3) == 2

assert Solution().minConnectedGroups(intervals = [[1,2],[2,3],[3,4],[4,5]], k = 1) == 1

assert Solution().minConnectedGroups(intervals = [[1,3],[4,7],[8,10],[15,20]], k = 5) == 2

assert Solution().minConnectedGroups(intervals = [[1,3],[7,10],[15,18],[20,25]], k = 6) == 3

assert Solution().minConnectedGroups(intervals = [[1,3],[3,5],[5,7],[7,9]], k = 2) == 1

assert Solution().minConnectedGroups(intervals = [[1, 1000000000]], k = 500000000) == 1

assert Solution().minConnectedGroups(intervals = [[1, 10], [15, 20], [25, 30]], k = 6) == 2

assert Solution().minConnectedGroups(intervals = [[1,3],[4,7],[8,11],[12,15]], k = 5) == 2

assert Solution().minConnectedGroups(intervals = [[1,10],[15,25],[30,40]], k = 9) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[2,3],[3,4]], k = 0) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[1,10]], k = 5) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[4,8],[10,15]], k = 2) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[4,5],[7,8]], k = 2) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20]], k = 4) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[3,4],[5,6]], k = 1) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[5,10],[1,1],[3,3]], k = 1) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[10,20],[20,30]], k = 5) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[1,100]], k = 50) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[2,4],[6,8],[10,12]], k = 5) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[5,6],[8,10]], k = 3) == 2

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

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [5, 7], [9, 11], [13, 15], [17, 19]], k = 2) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [5, 7], [9, 11], [13, 15], [17, 19], [21, 23], [25, 27], [29, 31], [33, 35]], k = 1) == 9

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,6],[7,9],[10,12],[13,15],[16,18]], k = 1) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,7],[8,10],[15,20]], k = 5) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,6],[7,9],[10,12],[13,15]], k = 2) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[7,10],[15,18],[20,25]], k = 6) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1, 1000000000], [2000000000, 3000000000]], k = 500000000) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [5, 6], [10, 12], [15, 17], [20, 22], [25, 27]], k = 3) == 5

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

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[5,7],[10,12],[15,17],[20,22]], k = 4) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [5, 6], [9, 10], [13, 14], [17, 18], [21, 22], [25, 26]], k = 4) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1,100],[200,300],[400,500],[600,700]], k = 150) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35]], k = 4) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,1000000000],[2000000000,3000000000],[4000000000,5000000000]], k = 500000000) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [6, 10], [15, 20], [25, 30], [35, 40]], k = 5) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[20,25],[30,35],[40,45],[50,55]], k = 10) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,100],[101,200],[201,300],[301,400],[401,500]], k = 99) == 4

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

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,7],[8,11],[12,15],[16,19],[20,23]], k = 3) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,12],[13,18],[19,25],[26,30]], k = 3) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 100], [200, 300], [400, 500], [600, 700], [800, 900]], k = 150) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [10, 15], [20, 25], [30, 35], [40, 45]], k = 5) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [4, 6], [7, 9], [10, 12], [13, 15]], k = 3) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,100],[200,300],[400,500],[600,700]], k = 100) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[4,6],[8,10],[12,14],[16,18]], k = 3) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [3, 5], [6, 8], [10, 12], [14, 16], [18, 20]], k = 2) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [4, 5], [7, 8], [10, 11], [13, 14]], k = 1) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[5,10],[15,20],[25,30],[35,40],[45,50]], k = 4) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[5,9],[9,13],[13,17],[17,21]], k = 3) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [10, 15], [20, 25], [30, 35], [40, 45], [50, 55], [60, 65], [70, 75], [80, 85], [90, 95]], k = 5) == 9

Answer: assert Solution().minConnectedGroups(intervals = [[1,100],[200,300],[400,500]], k = 100) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1, 50], [60, 100], [110, 150], [160, 200], [210, 250]], k = 45) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20],[21,25]], k = 8) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,7],[8,10],[11,15],[16,20]], k = 2) == 4

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

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

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [10, 15], [20, 25], [30, 35], [40, 45], [50, 55], [60, 65], [70, 75]], k = 8) == 7

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[3,5],[5,7],[7,9]], k = 2) == 1

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[3,5],[5,7],[7,9],[9,11]], k = 2) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [4, 5], [7, 8], [10, 11], [13, 14], [16, 17], [19, 20]], k = 2) == 6

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

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [5, 7], [9, 11], [13, 15], [17, 19], [21, 23], [25, 27], [29, 31]], k = 2) == 7

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[12,20],[22,30],[32,40],[42,50],[52,60]], k = 11) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [4, 5], [7, 8], [10, 11], [13, 14], [16, 17], [19, 20], [22, 23], [25, 26], [28, 29]], k = 1) == 10

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[15,20],[25,30],[35,40],[45,50]], k = 10) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 100], [200, 300], [400, 500], [600, 700], [800, 900]], k = 50) == 5

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

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,100],[200,300],[400,500],[600,700],[800,900],[1000,1100]], k = 150) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[20,30],[40,50],[60,70],[80,90]], k = 10) == 4

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35],[40,45],[50,55]], k = 9) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1, 10], [11, 20], [21, 30], [31, 40], [41, 50], [51, 60], [61, 70]], k = 10) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [10, 15], [20, 25], [30, 35], [40, 45]], k = 10) == 4

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[5,7],[9,11],[13,15],[17,19],[21,23]], k = 1) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1, 1000000000]], k = 500000000) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [10, 14], [19, 23], [28, 32], [37, 41]], k = 8) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[6,8],[11,13],[16,18],[21,23],[26,28]], k = 5) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [10, 15], [20, 25], [30, 35], [40, 45], [50, 55], [60, 65], [70, 75], [80, 85], [90, 95]], k = 10) == 9

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

Answer: assert Solution().minConnectedGroups(intervals = [[1, 10], [15, 20], [25, 30]], k = 6) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[2,5],[10,15],[20,25],[30,35]], k = 9) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20],[21,25]], k = 5) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[3,5],[7,10],[12,15],[17,20]], k = 2) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[15,20],[25,30],[35,40],[45,50],[55,60]], k = 14) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20],[21,25]], k = 3) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35]], k = 5) == 3

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35],[40,45]], k = 10) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[15,20],[25,30],[35,40]], k = 10) == 3

Answer: assert Solution().minConnectedGroups(intervals = [[1,20],[30,50],[60,80],[90,110]], k = 10) == 3

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

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [10, 15], [20, 25], [30, 35]], k = 5) == 3

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[11,20],[21,30],[31,40],[41,50]], k = 5) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,7],[8,11],[12,15]], k = 5) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[6,8],[11,13],[16,18],[21,23]], k = 5) == 4

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

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

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [4, 5], [7, 8], [10, 11], [13, 14], [16, 17], [19, 20], [22, 23], [25, 26], [28, 29]], k = 3) == 9

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,6],[7,9],[10,12],[13,15]], k = 3) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[20,30],[40,50],[60,70],[80,90]], k = 15) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[4,7],[8,10],[11,13],[14,17],[18,20]], k = 3) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[3,5],[6,8],[9,11],[12,14],[15,17]], k = 3) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[5, 10], [15, 20], [25, 30], [35, 40], [45, 50], [55, 60], [65, 70]], k = 5) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1, 10], [11, 20], [21, 30], [31, 40], [41, 50]], k = 10) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20],[21,25],[26,30]], k = 4) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20],[21,25],[26,30]], k = 5) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [6, 8], [10, 12], [15, 18], [20, 22]], k = 2) == 4

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35],[40,45],[50,55]], k = 20) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35],[40,45]], k = 1) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[20,30],[40,50],[60,70],[80,90],[100,110]], k = 9) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20],[21,25],[26,30],[31,35]], k = 5) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[5,7],[9,11],[13,15],[17,19]], k = 2) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,3],[5,8],[10,12],[15,17],[20,22]], k = 2) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[20,25],[35,40],[50,55],[60,65],[70,75]], k = 10) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[15,25],[30,40]], k = 9) == 2

Answer: assert Solution().minConnectedGroups(intervals = [[1, 10], [20, 30], [40, 50], [60, 70], [80, 90]], k = 15) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 3], [5, 7], [9, 11], [13, 15], [17, 19], [21, 23], [25, 27]], k = 3) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [6, 10], [11, 15], [16, 20], [21, 25], [26, 30], [31, 35], [36, 40], [41, 45], [46, 50]], k = 4) == 9

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [6, 10], [11, 15], [16, 20], [21, 25], [26, 30], [31, 35], [36, 40], [41, 45]], k = 10) == 7

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [4, 5], [7, 8], [10, 11], [13, 14], [16, 17], [19, 20], [22, 23], [25, 26], [28, 29]], k = 2) == 9

Answer: assert Solution().minConnectedGroups(intervals = [[1, 10], [20, 30], [40, 50], [60, 70], [80, 90]], k = 10) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[5,7],[9,12],[15,18],[21,23]], k = 4) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 10], [15, 20], [25, 30], [35, 40], [45, 50]], k = 5) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 2], [10, 11], [20, 21], [30, 31], [40, 41], [50, 51]], k = 8) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[15,20],[25,30],[35,40]], k = 20) == 2

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

Answer: assert Solution().minConnectedGroups(intervals = [[1,2],[2,3],[3,4],[4,5],[5,6],[6,7]], k = 1) == 1

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35],[40,45]], k = 3) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[10,15],[20,25],[30,35],[40,45],[50,55]], k = 3) == 6

Answer: assert Solution().minConnectedGroups(intervals = [[1,10],[20,30],[40,50],[60,70],[80,90],[100,110]], k = 15) == 5

Answer: assert Solution().minConnectedGroups(intervals = [[1,5],[6,10],[11,15],[16,20],[21,25]], k = 4) == 4

Answer: assert Solution().minConnectedGroups(intervals = [[1, 5], [6, 10], [11, 15], [16, 20], [21, 25], [26, 30], [31, 35], [36, 40], [41, 45], [46, 50]], k = 5) == 9

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 minConnectedGroups(self, intervals: List[List[int]], k: int) -> int:
        intervals.sort()
        merged = [intervals[0]]
        for s, e in intervals[1:]:
            if merged[-1][1] < s:
                merged.append([s, e])
            else:
                merged[-1][1] = max(merged[-1][1], e)
        ans = len(merged)
        for i, (_, e) in enumerate(merged):
            j = bisect_left(merged, [e + k + 1, 0])
            ans = min(ans, len(merged) - (j - i - 1))
        return ans