
def 							a		(				snake_case__:   int   ):


				'''simple docstring'''

				if not isinstance(snake_case__		,  snake_case__   ):
								raise ValueError('''multiplicative_persistence() only accepts integral values'''   )
				if num < 0:
								raise ValueError('''multiplicative_persistence() does not accept negative values'''   )

				lowercase_	    =   0
				lowercase_	    =   str(snake_case__   )

				while len(snake_case__   ) != 1:
								lowercase_	    =   [int(snake_case__   ) for i in num_string]

								lowercase_	    =   1
								for i in range(0		,  len(snake_case__   )   ):
												total *= numbers[i]

								lowercase_	    =   str(snake_case__   )

								steps += 1
				return steps




def 							a		(				snake_case__:   int   ):


				'''simple docstring'''

				if not isinstance(snake_case__		,  snake_case__   ):
								raise ValueError('''additive_persistence() only accepts integral values'''   )
				if num < 0:
								raise ValueError('''additive_persistence() does not accept negative values'''   )

				lowercase_	    =   0
				lowercase_	    =   str(snake_case__   )

				while len(snake_case__   ) != 1:
								lowercase_	    =   [int(snake_case__   ) for i in num_string]

								lowercase_	    =   0
								for i in range(0		,  len(snake_case__   )   ):
												total += numbers[i]

								lowercase_	    =   str(snake_case__   )

								steps += 1
				return steps


if __name__ == "__main__":
						import doctest

						doctest.testmod()











from __future__ import annotations
def      A__       (    SCREAMING_SNAKE_CASE_   :      int						)  ->  bool:






   """simple docstring"""







   _UpperCAmelCase           =   str(SCREAMING_SNAKE_CASE_						)
   return len(SCREAMING_SNAKE_CASE_						) == 9 and set(SCREAMING_SNAKE_CASE_						) == set('''123456789'''						)
def      A__       (    )  ->  int | None:






   """simple docstring"""







   for base_num in range(99_99				,	49_99				,	-1						):
      _UpperCAmelCase           =   10_00_02 * base_num
      if is_9_pandigital(SCREAMING_SNAKE_CASE_						):
         return candidate

   for base_num in range(3_33				,	99				,	-1						):
      _UpperCAmelCase           =   1_00_20_03 * base_num
      if is_9_pandigital(SCREAMING_SNAKE_CASE_						):
         return candidate

   return None


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