



class 						snake_case  :



   '''simple docstring'''






   def __init__(    self							:	Tuple ,						lowerCAmelCase							:	List[str] = "" ,						lowerCAmelCase							:	str = False) -> None:

    """simple docstring"""







    _snake_case			:     dict[str, RadixNode] 		=    {}

    # A node will be a leaf if the tree contains its word
    _snake_case			:     Union[str, Any] 		=    is_leaf

    _snake_case			:     Optional[Any] 		=    prefix

   def    UpperCamelCase_     (    self							:	List[Any] ,						lowerCAmelCase							:	List[str]) -> tuple[str, str, str]:

    """simple docstring"""







    _snake_case			:     Optional[int] 		=    0
    for q, w in zip(self.prefix ,						_SCREAMING_SNAKE_CASE):
     if q != w:
      break

     x += 1

    return self.prefix[:x], self.prefix[x:], word[x:]

   def    UpperCamelCase_     (    self							:	str ,						lowerCAmelCase							:	Optional[Any]) -> None:

    """simple docstring"""







    for word in words:
     self.insert(_SCREAMING_SNAKE_CASE)

   def    UpperCamelCase_     (    self							:	Dict ,						lowerCAmelCase							:	List[str]) -> None:

    """simple docstring"""







    if self.prefix == word:
     _snake_case			:     List[str] 		=    True

    # Case 2: The node has no edges that have a prefix to the word
    # Solution: We create an edge from the current node to a new one
    # containing the word
    elif word[0] not in self.nodes:
     _snake_case			:     Any 		=    RadixNode(prefix=_SCREAMING_SNAKE_CASE ,						is_leaf=_SCREAMING_SNAKE_CASE)

    else:
     _snake_case			:     Tuple 		=    self.nodes[word[0]]
     _snake_case			:     Any 		=    incoming_node.match(
         _SCREAMING_SNAKE_CASE)

     # Case 3: The node prefix is equal to the matching
     # Solution: We insert remaining word on the next node
     if remaining_prefix == "":
      self.nodes[matching_string[0]].insert(_SCREAMING_SNAKE_CASE)

     # Case 4: The word is greater equal to the matching
     # Solution: Create a node in between both nodes, change
     # prefixes and add the new node for the remaining word
     else:
      _snake_case			:     List[str] 		=    remaining_prefix

      _snake_case			:     Dict 		=    self.nodes[matching_string[0]]
      _snake_case			:     List[str] 		=    RadixNode(_SCREAMING_SNAKE_CASE ,						_SCREAMING_SNAKE_CASE)
      _snake_case			:     Union[str, Any] 		=    aux_node

      if remaining_word == "":
       _snake_case			:     Optional[Any] 		=    True
      else:
       self.nodes[matching_string[0]].insert(_SCREAMING_SNAKE_CASE)

   def    UpperCamelCase_     (    self							:	int ,						lowerCAmelCase							:	Tuple) -> bool:

    """simple docstring"""







    _snake_case			:     Any 		=    self.nodes.get(word[0] ,						_SCREAMING_SNAKE_CASE)
    if not incoming_node:
     return False
    else:
     _snake_case			:     str 		=    incoming_node.match(
         _SCREAMING_SNAKE_CASE)
     # If there is remaining prefix, the word can't be on the tree
     if remaining_prefix != "":
      return False
     # This applies when the word and the prefix are equal
     elif remaining_word == "":
      return incoming_node.is_leaf
     # We have word remaining so we check the next node
     else:
      return incoming_node.find(_SCREAMING_SNAKE_CASE)

   def    UpperCamelCase_     (    self							:	List[Any] ,						lowerCAmelCase							:	Union[str, Any]) -> bool:

    """simple docstring"""







    _snake_case			:     Optional[int] 		=    self.nodes.get(word[0] ,						_SCREAMING_SNAKE_CASE)
    if not incoming_node:
     return False
    else:
     _snake_case			:     Tuple 		=    incoming_node.match(
         _SCREAMING_SNAKE_CASE)
     # If there is remaining prefix, the word can't be on the tree
     if remaining_prefix != "":
      return False
     # We have word remaining so we check the next node
     elif remaining_word != "":
      return incoming_node.delete(_SCREAMING_SNAKE_CASE)
     else:
      # If it is not a leaf, we don't have to delete
      if not incoming_node.is_leaf:
       return False
      else:
       # We delete the nodes if no edges go from it
       if len(incoming_node.nodes) == 0:
        del self.nodes[word[0]]
        # We merge the current node with its only child
        if len(self.nodes) == 1 and not self.is_leaf:
         _snake_case			:     Optional[int] 		=    list(self.nodes.values())[0]
         _snake_case			:     Any 		=    merging_node.is_leaf
         self.prefix += merging_node.prefix
         _snake_case			:     Dict 		=    merging_node.nodes
          # If there is more than 1 edge, we just mark it as non-leaf
       elif len(incoming_node.nodes) > 1:
        _snake_case			:     Optional[Any] 		=    False
       # If there is 1 edge, we merge it with its child
       else:
        _snake_case			:     Dict 		=    list(incoming_node.nodes.values())[0]
        _snake_case			:     Optional[int] 		=    merging_node.is_leaf
        incoming_node.prefix += merging_node.prefix
        _snake_case			:     Optional[Any] 		=    merging_node.nodes

       return True


   def    UpperCamelCase_     (    self							:	Optional[int] ,						lowerCAmelCase							:	Union[str, Any] = 0) -> None:

    """simple docstring"""







    if self.prefix != "":
     print("""-""" * height ,						self.prefix ,						"""  (leaf)""" if self.is_leaf else """""")

    for value in self.nodes.values():
     value.print_tree(height + 1)



def 					lowercase				(      )     ->   Any:
 _snake_case			:     Any 		=    '''banana bananas bandana band apple all beast'''.split()
 _snake_case			:     List[str] 		=    RadixNode()
 root.insert_many(SCREAMING_SNAKE_CASE__ )

 assert all(root.find(SCREAMING_SNAKE_CASE__ ) for word in words )
 assert not root.find("""bandanas""" )
 assert not root.find("""apps""" )
 root.delete("""all""" )
 assert not root.find("""all""" )
 root.delete("""banana""" )
 assert not root.find("""banana""" )
 assert root.find("""bananas""" )

 return True



def 					lowercase				(      )     ->   Dict:
 assert test_trie()



def 					lowercase				(      )     ->   Any:
 _snake_case			:     str 		=    RadixNode()
 _snake_case			:     Dict 		=    '''banana bananas bandanas bandana band apple all beast'''.split()
 root.insert_many(SCREAMING_SNAKE_CASE__ )

 print("""Words:""" ,				SCREAMING_SNAKE_CASE__ )
 print("""Tree:""" )
 root.print_tree()


if __name__ == "__main__":
  main()




from math import isclose, sqrt


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       ):
  A_      :				List[str]		     =					point_y / 4 / point_x
  A_      :				Union[str, Any]		     =					2 * normal_gradient / (1 + normal_gradient * normal_gradient)
  A_      :				int		     =					(1 - normal_gradient * normal_gradient) / (
      1 + normal_gradient * normal_gradient
  )
  A_      :				List[str]		     =					(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

  # to find the next point, solve the simultaeneous equations:
  # y^2 + 4x^2 = 100
  # y - b = m * (x - a)
  # ==> A x^2 + B x + C = 0
  A_      :				List[str]		     =					outgoing_gradient**2 + 4
  A_      :				List[str]		     =					2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
  A_      :				Any		     =					(point_y - outgoing_gradient * point_x) ** 2 - 100

  A_      :				str		     =					(
      -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
  ) / (2 * quadratic_term)
  A_      :				Any		     =					(
      -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term       )
  ) / (2 * quadratic_term)

  # two solutions, one of which is our input point
  A_      :				int		     =					x_minus if isclose(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       ) else x_plus
  A_      :				Any		     =					point_y + outgoing_gradient * (next_x - point_x)

  return next_x, next_y, outgoing_gradient


def   _SCREAMING_SNAKE_CASE	( SCREAMING_SNAKE_CASE = 1.4 ,			SCREAMING_SNAKE_CASE = -9.6       ):
  A_      :				int		     =					0
  A_      :				float		     =					first_x_coord
  A_      :				float		     =					first_y_coord
  A_      :				float		     =					(1_0.1 - point_y) / (0.0 - point_x)

  while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
    A_     ,       A_     ,       A_      :				List[str]		     =					next_point(SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE ,			SCREAMING_SNAKE_CASE       )
    num_reflections += 1

  return num_reflections


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




