


class  lowerCamelCase__  :





     def __init__(self						,	UpperCAmelCase			)	->  Optional[int]:
      _lowercase				 =data
      _lowercase				 =None
      _lowercase				 =None





     def __str__(self			)	->  Optional[Any]:
      return f"{self.data}"







class  lowerCamelCase__  :





     def __init__(self			)	->  str:
      _lowercase				 =None
      _lowercase				 =None





     def __iter__(self			)	->  Dict:
      _lowercase				 =self.head
      while node:
       yield node.data
       _lowercase				 =node.next





     def __str__(self			)	->  Optional[Any]:
      return "->".join([str(__snake_case			) for item in self]			)





     def __len__(self			)	->  Optional[Any]:
      return sum(1 for _ in self			)





     def 	__A   (self						,	UpperCAmelCase			)	->  Tuple:
      self.insert_at_nth(0						,	__snake_case			)





     def 	__A   (self						,	UpperCAmelCase			)	->  Dict:
      self.insert_at_nth(len(self			)						,	__snake_case			)





     def 	__A   (self						,	UpperCAmelCase						,	UpperCAmelCase			)	->  Optional[int]:
      _lowercase				 =len(self			)

      if not 0 <= index <= length:
       raise IndexError('''list index out of range'''			)
      _lowercase				 =Node(__snake_case			)
      if self.head is None:
       _lowercase				 =new_node
      elif index == 0:
       _lowercase				 =new_node
       _lowercase				 =self.head
       _lowercase				 =new_node
      elif index == length:
       _lowercase				 =new_node
       _lowercase				 =self.tail
       _lowercase				 =new_node
      else:
       _lowercase				 =self.head
       for _ in range(0						,	__snake_case			):
        _lowercase				 =temp.next
       _lowercase				 =new_node
       _lowercase				 =temp.previous
       _lowercase				 =temp
       _lowercase				 =new_node





     def 	__A   (self			)	->  Tuple:
      return self.delete_at_nth(0			)





     def 	__A   (self			)	->  str:
      return self.delete_at_nth(len(self			) - 1			)





     def 	__A   (self						,	UpperCAmelCase			)	->  Optional[Any]:
      _lowercase				 =len(self			)

      if not 0 <= index <= length - 1:
       raise IndexError('''list index out of range'''			)
      _lowercase				 =self.head  # default first node
      if length == 1:
       _lowercase				 =None
      elif index == 0:
       _lowercase				 =self.head.next
       _lowercase				 =None
      elif index == length - 1:
       _lowercase				 =self.tail
       _lowercase				 =self.tail.previous
       _lowercase				 =None
      else:
       _lowercase				 =self.head
       for _ in range(0						,	__snake_case			):
        _lowercase				 =temp.next
       _lowercase				 =temp
       _lowercase				 =temp.previous
       _lowercase				 =temp.next
      return delete_node.data





     def 	__A   (self						,	UpperCAmelCase			)	->  Dict:
      _lowercase				 =self.head

      while current.data != data:  # Find the position to delete
       if current.next:
        _lowercase				 =current.next
       else:  # We have reached the end an no value matches
        raise ValueError('''No data matching given value'''			)

      if current == self.head:
       self.delete_head()

      elif current == self.tail:
       self.delete_tail()

      else:  # Before: 1 <--> 2(current) <--> 3
       _lowercase				 =current.next  # 1 --> 3
       _lowercase				 =current.previous  # 1 <--> 3
      return data





     def 	__A   (self			)	->  Any:
      return len(self			) == 0



def 							UpperCAmelCase_   (				)	->     None:







 """simple docstring"""


 _lowercase				 =DoublyLinkedList()
 assert linked_list.is_empty() is True
 assert str(UpperCAmelCase_		) == ""

 try:
  linked_list.delete_head()
  raise AssertionError  # This should not happen.
 except IndexError:
  assert True  # This should happen.

 try:
  linked_list.delete_tail()
  raise AssertionError  # This should not happen.
 except IndexError:
  assert True  # This should happen.

 for i in range(10		):
  assert len(UpperCAmelCase_		) == i
  linked_list.insert_at_nth(UpperCAmelCase_	,		i + 1		)
 assert str(UpperCAmelCase_		) == "->".join(str(UpperCAmelCase_		) for i in range(1	,		11		)		)

 linked_list.insert_at_head(0		)
 linked_list.insert_at_tail(11		)
 assert str(UpperCAmelCase_		) == "->".join(str(UpperCAmelCase_		) for i in range(0	,		12		)		)

 assert linked_list.delete_head() == 0
 assert linked_list.delete_at_nth(9		) == 10
 assert linked_list.delete_tail() == 11
 assert len(UpperCAmelCase_		) == 9
 assert str(UpperCAmelCase_		) == "->".join(str(UpperCAmelCase_		) for i in range(1	,		10		)		)


if __name__ == "__main__":
  from doctest import testmod

  testmod()




from __future__ import annotations






def        lowerCamelCase_  (				UpperCAmelCase_				:  int						)    ->					list[int]:




	'''simple docstring'''




	_UpperCamelCase :		Any								=    [0] * max_one

	for i in range(0				,							UpperCAmelCase_						):
		_UpperCamelCase :		List[Any]								=    i

	for i in range(2				,							UpperCAmelCase_						):
		if totients[i] == i:
			for j in range(UpperCAmelCase_				,							UpperCAmelCase_				,							UpperCAmelCase_						):
				totients[j] -= totients[j] // i

	return totients






def        lowerCamelCase_  (				UpperCAmelCase_				:  int				,							UpperCAmelCase_				:  int						)    ->					bool:




	'''simple docstring'''




	return sorted(str(UpperCAmelCase_						)						) == sorted(str(UpperCAmelCase_						)						)






def        lowerCamelCase_  (				UpperCAmelCase_				:  int = 1_0_0_0_0_0_0_0						)    ->					int:




	'''simple docstring'''




	_UpperCamelCase :		List[Any]								=    1  # i
	_UpperCamelCase :		List[str]								=    0  # φ(i)
	_UpperCamelCase :		str								=    get_totients(max_n + 1						)

	for i in range(2				,							max_n + 1						):
		_UpperCamelCase :		List[str]								=    totients[i]

		if i * min_denominator < min_numerator * t and has_same_digits(UpperCAmelCase_				,							UpperCAmelCase_						):
			_UpperCamelCase :		Optional[int]								=    i
			_UpperCamelCase :		Tuple								=    t

	return min_numerator


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




