
_A	  =					{
    '''0''': 1,
    '''1''': 1,
    '''2''': 2,
    '''3''': 6,
    '''4''': 24,
    '''5''': 120,
    '''6''': 720,
    '''7''': 5_040,
    '''8''': 40_320,
    '''9''': 362_880,
}

_A	  =					{145: 145}

_A	  =					{
    145: 0,
    169: 3,
    36_301: 3,
    1_454: 3,
    871: 2,
    45_361: 2,
    872: 2,
}




def 				lowerCamelCase__				(						a__  :  int     )		->	int:
       if n in CACHE_SUM_DIGIT_FACTORIALS:
              return CACHE_SUM_DIGIT_FACTORIALS[n]
       UpperCamelCase_							       =       sum(DIGIT_FACTORIALS[let] for let in str(a__     )     )
       UpperCamelCase_							       =       ret
       return ret




def 				lowerCamelCase__				(						a__  :  int	,      a__  :  set | None = None     )		->	int:
       UpperCamelCase_							       =       previous or set()
       if n in CHAIN_LENGTH_CACHE:
              return CHAIN_LENGTH_CACHE[n]
       UpperCamelCase_							       =       sum_digit_factorials(a__     )
       if next_number in previous:
              UpperCamelCase_							       =       0
              return 0
       else:
              previous.add(a__     )
              UpperCamelCase_							       =       1 + chain_length(a__	,      a__     )
              UpperCamelCase_							       =       ret
              return ret




def 				lowerCamelCase__				(						a__  :  int = 60	,      a__  :  int = 100_0000     )		->	int:
       return sum(1 for i in range(1	,      a__     ) if chain_length(a__     ) == num_terms     )


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





def 				lowerCamelCase__				(						a__  :  dict	,      a__  :  int     )		->	Optional[Any]:
       if not root or root not in binary_tree_mirror_dictionary:
              return
       UpperCamelCase_       ,				UpperCamelCase_							       =       binary_tree_mirror_dictionary[root][:2]
       UpperCamelCase_							       =       [right_child, left_child]
       binary_tree_mirror_dict(a__	,      a__     )
       binary_tree_mirror_dict(a__	,      a__     )




def 				lowerCamelCase__				(						a__  :  dict	,      a__  :  int = 1     )		->	dict:
       if not binary_tree:
              raise ValueError("""binary tree cannot be empty"""     )
       if root not in binary_tree:
              UpperCamelCase_							       =       f'''root {root} is not present in the binary_tree'''
              raise ValueError(a__     )
       UpperCamelCase_							       =       dict(a__     )
       binary_tree_mirror_dict(a__	,      a__     )
       return binary_tree_mirror_dictionary


if __name__ == "__main__":
      _A	  =					{1: [2, 3], 2: [4, 5], 3: [6, 7], 7: [8, 9]}
      print(F'''Binary tree: {binary_tree}''')
      _A	  =					binary_tree_mirror(binary_tree, 5)
      print(F'''Binary tree mirror: {binary_tree_mirror_dictionary}''')



