
'''simple docstring'''
import math







def        A_     (						__SCREAMING_SNAKE_CASE						:  int )				->       bool:





  """simple docstring"""

  if 1 < number < 4:
    # 2 and 3 are primes
    return True
  elif number < 2 or number % 2 == 0 or number % 3 == 0:
    # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
    return False

  # All primes number are in format of 6k +/- 1
  for i in range(5			,     int(math.sqrt(__SCREAMING_SNAKE_CASE ) + 1 )			,     6 ):
    if number % i == 0 or number % (i + 2) == 0:
      return False
  return True







def        A_     (						__SCREAMING_SNAKE_CASE						:  float = 0.1 )				->       int:





  """simple docstring"""

  __A		:     str												=		3
  __A		:     Optional[Any]												=		3

  while primes / (2 * j - 1) >= ratio:
    for i in range(j * j + j + 1			,     (j + 2) * (j + 2)			,     j + 1 ):
      primes += is_prime(__SCREAMING_SNAKE_CASE )
    j += 2
  return j


if __name__ == "__main__":
     import doctest

     doctest.testmod()





'''simple docstring'''
def        A_     (						__SCREAMING_SNAKE_CASE						:  Dict )				->       Optional[Any]:





  """simple docstring"""

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







def        A_     (						__SCREAMING_SNAKE_CASE						:  Union[str, Any] )				->       Optional[int]:





  """simple docstring"""

  if not head or not head.next:
    return True

  # 1. Get the midpoint (slow)
  __A		:     int												=		head
  while fast and fast.next:
    __A							,					__A		:     Optional[int]												=		fast.next.next, slow.next

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

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

  return True







def        A_     (						__SCREAMING_SNAKE_CASE						:  Dict )				->       Any:





  """simple docstring"""

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



