


class       lowerCamelCase__   :





     def __init__(self		,     UpperCAmelCase = ""		,     UpperCAmelCase = False       )     ->   None:
      _lowercase							      ={}

      # A node will be a leaf if the tree contains its word
      _lowercase							      =is_leaf

      _lowercase							      =prefix





     def      __A					(self		,     UpperCAmelCase       )     ->   tuple[str, str, str]:
      _lowercase							      =0
      for q, w in zip(self.prefix		,     UpperCAmelCase_       ):
       if q != w:
        break

       x += 1

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





     def      __A					(self		,     UpperCAmelCase       )     ->   None:



      for word in words:
       self.insert(UpperCAmelCase_       )





     def      __A					(self		,     UpperCAmelCase       )     ->   None:



      if self.prefix == word:
       _lowercase							      =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:
       _lowercase							      =RadixNode(prefix=UpperCAmelCase_		,     is_leaf=UpperCAmelCase_       )

      else:
       _lowercase							      =self.nodes[word[0]]
       _lowercase							      =incoming_node.match(
           UpperCAmelCase_       )

       # 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(UpperCAmelCase_       )

       # 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:
        _lowercase							      =remaining_prefix

        _lowercase							      =self.nodes[matching_string[0]]
        _lowercase							      =RadixNode(UpperCAmelCase_		,     UpperCAmelCase_       )
        _lowercase							      =aux_node

        if remaining_word == "":
         _lowercase							      =True
        else:
         self.nodes[matching_string[0]].insert(UpperCAmelCase_       )





     def      __A					(self		,     UpperCAmelCase       )     ->   bool:
      _lowercase							      =self.nodes.get(word[0]		,     UpperCAmelCase_       )
      if not incoming_node:
       return False
      else:
       _lowercase							      =incoming_node.match(
           UpperCAmelCase_       )
       # 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(UpperCAmelCase_       )





     def      __A					(self		,     UpperCAmelCase       )     ->   bool:
      _lowercase							      =self.nodes.get(word[0]		,     UpperCAmelCase_       )
      if not incoming_node:
       return False
      else:
       _lowercase							      =incoming_node.match(
           UpperCAmelCase_       )
       # 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(UpperCAmelCase_       )
       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:
           _lowercase							      =list(self.nodes.values()       )[0]
           _lowercase							      =merging_node.is_leaf
           self.prefix += merging_node.prefix
           _lowercase							      =merging_node.nodes
                  # If there is more than 1 edge, we just mark it as non-leaf
         elif len(incoming_node.nodes       ) > 1:
          _lowercase							      =False
         # If there is 1 edge, we merge it with its child
         else:
          _lowercase							      =list(incoming_node.nodes.values()       )[0]
          _lowercase							      =merging_node.is_leaf
          incoming_node.prefix += merging_node.prefix
          _lowercase							      =merging_node.nodes

         return True





     def      __A					(self		,     UpperCAmelCase = 0       )     ->   None:



      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 				UpperCAmelCase_					(							)     ->			bool:







 """simple docstring"""


 _lowercase							      ="banana bananas bandana band apple all beast".split()
 _lowercase							      =RadixNode()
 root.insert_many(__a						)

 assert all(root.find(__a						) 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 				UpperCAmelCase_					(							)     ->			None:







 """simple docstring"""


 assert test_trie()
def 				UpperCAmelCase_					(							)     ->			None:







 """simple docstring"""


 _lowercase							      =RadixNode()
 _lowercase							      ="banana bananas bandanas bandana band apple all beast".split()
 root.insert_many(__a						)

 print('''Words:'''     ,						__a						)
 print('''Tree:'''						)
 root.print_tree()


if __name__ == "__main__":
  main()








from math import ceil, sqrt





def      lowerCAmelCase_ (       __a = 1000000		)     ->		int:







  """simple docstring"""





  lowerCamelCase__:      Optional[int]			  =0

  for outer_width in range(3	,    (limit // 4) + 2		):
    if outer_width**2 > limit:
      lowerCamelCase__:      Dict			  =max(ceil(sqrt(outer_width**2 - limit		)		)	,    1		)
    else:
      lowerCamelCase__:      str			  =1
    if (outer_width - hole_width_lower_bound) % 2:
      hole_width_lower_bound += 1

    answer += (outer_width - hole_width_lower_bound - 2) // 2 + 1

  return answer


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

