



"""simple docstring"""




import sys

def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int		,						SCREAMING_SNAKE_CASE__			:		list[int]       )						->     int:





     '''simple docstring'''

     if not numbers:
          return 0
     if target == 0 and target in numbers:
          return 0
     if not isinstance(SCREAMING_SNAKE_CASE__		,						(list, tuple)       ) or not all(
         isinstance(SCREAMING_SNAKE_CASE__		,						SCREAMING_SNAKE_CASE__       ) for number in numbers       ):
          raise ValueError("numbers must be an iterable of integers"       )

     _UpperCAmelCase    :			List[Any]              =					0
     _UpperCAmelCase    :			str              =					sys.maxsize

     while right < len(SCREAMING_SNAKE_CASE__       ):
          curr_sum += numbers[right]
          while curr_sum >= target and left <= right:
               _UpperCAmelCase    :			Any              =					min(SCREAMING_SNAKE_CASE__		,						right - left + 1       )
               curr_sum -= numbers[left]
               left += 1
          right += 1

     return 0 if min_len == sys.maxsize else min_len





"""simple docstring"""




_lowerCAmelCase       :      str							=							{
    "0": 1,
    "1": 1,
    "2": 2,
    "3": 6,
    "4": 24,
    "5": 1_20,
    "6": 7_20,
    "7": 50_40,
    "8": 4_03_20,
    "9": 36_28_80,
}

_lowerCAmelCase       :      Optional[int]							=							{1_45: 1_45}

_lowerCAmelCase       :      int							=							{
    1_45: 0,
    1_69: 3,
    3_63_01: 3,
    14_54: 3,
    8_71: 2,
    4_53_61: 2,
    8_72: 2,
}

def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int       )						->     int:





     '''simple docstring'''

     if n in CACHE_SUM_DIGIT_FACTORIALS:
          return CACHE_SUM_DIGIT_FACTORIALS[n]
     _UpperCAmelCase    :			Dict              =					sum(DIGIT_FACTORIALS[let] for let in str(SCREAMING_SNAKE_CASE__       )       )
     _UpperCAmelCase    :			Any              =					ret
     return ret

def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int		,						SCREAMING_SNAKE_CASE__			:		set | None = None       )						->     int:





     '''simple docstring'''

     _UpperCAmelCase    :			Dict              =					previous or set()
     if n in CHAIN_LENGTH_CACHE:
          return CHAIN_LENGTH_CACHE[n]
     _UpperCAmelCase    :			Dict              =					sum_digit_factorials(SCREAMING_SNAKE_CASE__       )
     if next_number in previous:
          _UpperCAmelCase    :			List[str]              =					0
          return 0
     else:
          previous.add(SCREAMING_SNAKE_CASE__       )
          _UpperCAmelCase    :			str              =					1 + chain_length(SCREAMING_SNAKE_CASE__		,						SCREAMING_SNAKE_CASE__       )
          _UpperCAmelCase    :			str              =					ret
          return ret

def 		__snake_case		(   SCREAMING_SNAKE_CASE__			:		int = 60		,						SCREAMING_SNAKE_CASE__			:		int = 1_000_000       )						->     int:





     '''simple docstring'''

     return sum(1 for i in range(1		,						SCREAMING_SNAKE_CASE__       ) if chain_length(SCREAMING_SNAKE_CASE__       ) == num_terms       )


if __name__ == "__main__":
 print(F"{solution() = }")
