




"""simple docstring"""







def 							lowercase    (snake_case__      :       int = 1_000_000       )   ->     Dict:


				'''simple docstring'''






				lowerCAmelCase              = set(range(3					,				__snake_case					,				2       )       )
				primes.add(2       )
				for p in range(3					,				__snake_case					,				2       ):
								if p not in primes:
												continue
								primes.difference_update(set(range(p * p					,				__snake_case					,				__snake_case       )       )       )

				lowerCAmelCase              = [float(__snake_case       ) for n in range(limit + 1       )]

				for p in primes:
								for n in range(__snake_case					,				limit + 1					,				__snake_case       ):
												phi[n] *= 1 - 1 / p

				return int(sum(phi[2:]       )       )


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













def      lowercase__      (     __snake_case    :							Dict  ):


					'''simple docstring'''


					if not head:
										return True
					# split the list to two parts
					UpperCAmelCase_       ,  UpperCAmelCase_    :      Any            =						head.next, head
					while fast and fast.next:
										UpperCAmelCase_    :      str            =						fast.next.next
										UpperCAmelCase_    :      Union[str, Any]            =						slow.next
					UpperCAmelCase_    :      int            =						slow.next
					UpperCAmelCase_    :      List[Any]            =						None  # Don't forget here! But forget still works!
					# reverse the second part
					UpperCAmelCase_    :      Tuple            =						None
					while second:
										UpperCAmelCase_    :      int            =						second.next
										UpperCAmelCase_    :      Any            =						node
										UpperCAmelCase_    :      Optional[Any]            =						second
										UpperCAmelCase_    :      Tuple            =						nxt
					# compare two parts
					# second part has the same or one less node
					while node:
										if node.val != head.val:
															return False
										UpperCAmelCase_    :      Optional[Any]            =						node.next
										UpperCAmelCase_    :      Dict            =						head.next
					return True




def      lowercase__      (     __snake_case    :							Union[str, Any]  ):


					'''simple docstring'''


					if not head or not head.next:
										return True

					# 1. Get the midpoint (slow)
					UpperCAmelCase_    :      Any            =						head
					while fast and fast.next:
										UpperCAmelCase_       ,  UpperCAmelCase_    :      Tuple            =						fast.next.next, slow.next

					# 2. Push the second half into the stack
					UpperCAmelCase_    :      List[str]            =						[slow.val]
					while slow.next:
										UpperCAmelCase_    :      List[str]            =						slow.next
										stack.append(slow.val  )

					# 3. Comparison
					while stack:
										if stack.pop() != cur.val:
															return False
										UpperCAmelCase_    :      int            =						cur.next

					return True






def      lowercase__      (     __snake_case    :							Dict  ):


					'''simple docstring'''


					if not head or not head.next:
										return True
					UpperCAmelCase_    :      Tuple            =						{}
					UpperCAmelCase_    :      int            =						0
					while head:
										if head.val in d:
															d[head.val].append(__snake_case  )
										else:
															UpperCAmelCase_    :      List[Any]            =						[pos]
										UpperCAmelCase_    :      Any            =						head.next
										pos += 1
					UpperCAmelCase_    :      Dict            =						pos - 1
					UpperCAmelCase_    :      Optional[int]            =						0
					for v in d.values():
										if len(__snake_case  ) % 2 != 0:
															middle += 1
										else:
															UpperCAmelCase_    :      int            =						0
															for i in range(0     ,       len(__snake_case  )  ):
																				if v[i] + v[len(__snake_case  ) - 1 - step] != checksum:
																									return False
																				step += 1
										if middle > 1:
															return False
					return True






