





'''simple docstring'''






from __future__ import annotations

from random import random




class  UpperCAmelCase     :




      def __init__(			self			: Any			,			__snake_case			: int | None = None  )				->      str:
            _lowerCAmelCase							     =							value
            _lowerCAmelCase							     =							random()
            _lowerCAmelCase							     =							None
            _lowerCAmelCase							     =							None




      def __repr__(			self			: Dict  )				->      str:
            from pprint import pformat

            if self.left is None and self.right is None:
                  return f"'{self.value}: {self.prior:.5}'"
            else:
                  return pformat(
                      {f"{self.value}: {self.prior:.5}": (self.left, self.right)}			,			indent=1  )




      def __str__(			self			: Dict  )				->      str:
            _lowerCAmelCase							     =							str(self.value  ) + """ """
            _lowerCAmelCase							     =							str(self.left or """"""  )
            _lowerCAmelCase							     =							str(self.right or """"""  )
            return value + left + right




def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase  ):





      """simple docstring"""


      if root is None:  # None tree is split into 2 Nones
            return None, None
      elif root.value is None:
            return None, None
      else:
            if value < root.value:
                  _lowerCAmelCase     ,							_lowerCAmelCase							     =							split(root.left							,	lowerCAmelCase  )
                  return left, root
            else:
                  _lowerCAmelCase     ,							_lowerCAmelCase							     =							split(root.right							,	lowerCAmelCase  )
                  return root, right
def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase  ):





      """simple docstring"""


      if (not left) or (not right):  # If one node is None, return the other
            return left or right
      elif left.prior < right.prior:
            _lowerCAmelCase							     =							merge(left.right							,	lowerCAmelCase  )
            return left
      else:
            _lowerCAmelCase							     =							merge(lowerCAmelCase							,	right.left  )
            return right
def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase  ):





      """simple docstring"""


      _lowerCAmelCase							     =							Node(lowerCAmelCase  )
      _lowerCAmelCase     ,							_lowerCAmelCase							     =							split(lowerCAmelCase							,	lowerCAmelCase  )
      return merge(merge(lowerCAmelCase							,	lowerCAmelCase  )							,	lowerCAmelCase  )
def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase  ):





      """simple docstring"""


      _lowerCAmelCase     ,							_lowerCAmelCase							     =							split(lowerCAmelCase							,	value - 1  )
      _lowerCAmelCase     ,							_lowerCAmelCase							     =							split(lowerCAmelCase							,	lowerCAmelCase  )
      return merge(lowerCAmelCase							,	lowerCAmelCase  )
def    UpperCamelCase__      (		lowerCAmelCase  ):





      """simple docstring"""


      if not root:  # None
            return
      else:
            inorder(root.left  )
            print(root.value							,	end=""","""  )
            inorder(root.right  )
def    UpperCamelCase__      (		lowerCAmelCase							,	lowerCAmelCase  ):





      """simple docstring"""


      for arg in args.split():
            if arg[0] == "+":
                  _lowerCAmelCase							     =							insert(lowerCAmelCase							,	int(arg[1:]  )  )

            elif arg[0] == "-":
                  _lowerCAmelCase							     =							erase(lowerCAmelCase							,	int(arg[1:]  )  )

            else:
                  print("""Unknown command"""  )

      return root
def    UpperCamelCase__      (		):





      """simple docstring"""


      _lowerCAmelCase							     =							None
      print(
          """enter numbers to create a tree, + value to add value into treap, """
          """- value to erase all nodes with value. 'q' to quit. """  )

      _lowerCAmelCase							     =							input()
      while args != "q":
            _lowerCAmelCase							     =							interact_treap(lowerCAmelCase							,	lowerCAmelCase  )
            print(lowerCAmelCase  )
            _lowerCAmelCase							     =							input()

      print("""good by!"""  )


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      main()














def    SCREAMING_SNAKE_CASE_							(							__magic_name__					:  list[list[int]]	,			__magic_name__					:  int	,			__magic_name__					:  int	,			__magic_name__					:  list[int]					)		->			bool:






	"""simple docstring"""



	if graph[path[curr_ind - 1]][next_ver] == 0:
		return False

	# 2. Validate that next vertex is not already in path
	return not any(vertex == next_ver for vertex in path					)

def    SCREAMING_SNAKE_CASE_							(							__magic_name__					:  list[list[int]]	,			__magic_name__					:  list[int]	,			__magic_name__					:  int					)		->			bool:






	"""simple docstring"""



	if curr_ind == len(__magic_name__					):
		# return whether path exists between current and starting vertices
		return graph[path[curr_ind - 1]][path[0]] == 1

	# Recursive Step
	for next_ver in range(0	,			len(__magic_name__					)					):
		if valid_connection(__magic_name__	,			__magic_name__	,			__magic_name__	,			__magic_name__					):
			# Insert current vertex  into path as next transition
			UpperCamelCase     :str          =	next_ver
			# Validate created path
			if util_hamilton_cycle(__magic_name__	,			__magic_name__	,			curr_ind + 1					):
				return True
			# Backtrack
			UpperCamelCase     :Union[str, Any]          =	-1
	return False

def    SCREAMING_SNAKE_CASE_							(							__magic_name__					:  list[list[int]]	,			__magic_name__					:  int = 0					)		->			list[int]:






	"""simple docstring"""



	UpperCamelCase     :Union[str, Any]          =	[-1] * (len(__magic_name__					) + 1)
	# initialize start and end of path with starting index
	UpperCamelCase     :Any          =	start_index
	# evaluate and if we find answer return path either return empty array
	return path if util_hamilton_cycle(__magic_name__	,			__magic_name__	,			1					) else []
