



'''simple docstring'''

from __future__ import annotations

from random import random






class 						a_		:


							def __init__(				self   :						Optional[int]      ,						a_   :						int | None = None			)			->					List[Any]:
										snake_case:    Optional[int]      							=value
										snake_case:    Union[str, Any]      							=random()
										snake_case:    Node | None      							=None
										snake_case:    Node | None      							=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   :						Optional[Any]			)			->					str:
										snake_case:    List[str]      							=str(self.value			) + ' '
										snake_case:    int      							=str(self.left or ''			)
										snake_case:    Any      							=str(self.right or ''			)
										return value + left + right







def a_     (    __UpperCAmelCase				,							__UpperCAmelCase					)				->					tuple[Node | None, Node | None]:





			"""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:
									snake_case							,  snake_case:    Tuple      							=split(root.left				,							__UpperCAmelCase					)
									return left, root
						else:
									snake_case							,  snake_case:    Union[str, Any]      							=split(root.right				,							__UpperCAmelCase					)
									return root, right







def a_     (    __UpperCAmelCase				,							__UpperCAmelCase					)				->					Node | None:





			"""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:
						snake_case:    Tuple      							=merge(left.right				,							__UpperCAmelCase					)
						return left
			else:
						snake_case:    Union[str, Any]      							=merge(__UpperCAmelCase				,							right.left					)
						return right







def a_     (    __UpperCAmelCase				,							__UpperCAmelCase					)				->					Node | None:





			"""simple docstring"""







			snake_case:    Tuple      							=Node(__UpperCAmelCase					)
			snake_case							,  snake_case:    Any      							=split(__UpperCAmelCase				,							__UpperCAmelCase					)
			return merge(merge(__UpperCAmelCase				,							__UpperCAmelCase					)				,							__UpperCAmelCase					)







def a_     (    __UpperCAmelCase				,							__UpperCAmelCase					)				->					Node | None:





			"""simple docstring"""







			snake_case							,  snake_case:    Optional[int]      							=split(__UpperCAmelCase				,							value - 1					)
			snake_case							,  snake_case:    List[str]      							=split(__UpperCAmelCase				,							__UpperCAmelCase					)
			return merge(__UpperCAmelCase				,							__UpperCAmelCase					)







def a_     (    __UpperCAmelCase					)				->					None:





			"""simple docstring"""







			if not root:  # None
						return
			else:
						inorder(root.left					)
						print(root.value				,							end=','					)
						inorder(root.right					)







def a_     (    __UpperCAmelCase				,							__UpperCAmelCase					)				->					Node | None:





			"""simple docstring"""







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

						elif arg[0] == "-":
									snake_case:    int      							=erase(__UpperCAmelCase				,							int(arg[1:]					)					)

						else:
									print('Unknown command'					)

			return root







def a_     (    )				->					None:





			"""simple docstring"""







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

			snake_case:    List[Any]      							=input()
			while args != "q":
						snake_case:    str      							=interact_treap(__UpperCAmelCase				,							__UpperCAmelCase					)
						print(__UpperCAmelCase					)
						snake_case:    List[str]      							=input()

			print('good by!'					)


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	main()









'''simple docstring'''

def a_     (    __UpperCAmelCase				,							__UpperCAmelCase				,							__UpperCAmelCase					)				->					list:





			"""simple docstring"""







			snake_case:    Optional[int]      							=len(__UpperCAmelCase					)
			snake_case:    Optional[int]      							=[[0] * n for i in range(__UpperCAmelCase					)]
			for i in range(__UpperCAmelCase					):
						snake_case:    Optional[int]      							=y_points[i]

			for i in range(2				,							__UpperCAmelCase					):
						for j in range(__UpperCAmelCase				,							__UpperCAmelCase					):
									snake_case:    List[str]      							=(
									    (xa - x_points[j - i + 1]) * q[j][i - 1]
									    - (xa - x_points[j]) * q[j - 1][i - 1]
									) / (x_points[j] - x_points[j - i + 1])

			return [q[n - 1][n - 1], q]


if __name__ == "__main__":
	import doctest

	doctest.testmod()




