
'''simple docstring'''

from numpy import exp, pi, sqrt






def 	_snake_case						(      A_       :    List[str]     ,		A_       :    float = 0.0     ,		A_       :    float = 1.0   ):
  """simple docstring"""







  return 1 / sqrt(2 * pi * sigma**2   ) * exp(-((x - mu) ** 2) / (2 * sigma**2)   )


if __name__ == "__main__":
     import doctest

     doctest.testmod()




'''simple docstring'''

def 	_snake_case						(      A_       :    Optional[int]   ):
  """simple docstring"""







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






def 	_snake_case						(      A_       :    List[str]   ):
  """simple docstring"""







  if not head or not head.next:
    return True

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

  # 2. Push the second half into the stack
  a_      :	List[str]      =							[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_      :	Union[str, Any]      =							cur.next

  return True






def 	_snake_case						(      A_       :    Optional[int]   ):
  """simple docstring"""







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


