






"""simple docstring"""
from __future__ import annotations

from random import random







class 			UpperCamelCase_       :





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





def 				snake_case      (   A__	,A__						):
			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:
									UpperCAmelCase_     ,	UpperCAmelCase_     : Union[str, Any]			     =					split(root.left	,A__						)
									return left, root
						else:
									UpperCAmelCase_     ,	UpperCAmelCase_     : Tuple			     =					split(root.right	,A__						)
									return root, right

def 				snake_case      (   A__	,A__						):
			if (not left) or (not right):  # If one node is None, return the other
						return left or right
			elif left.prior < right.prior:
						UpperCAmelCase_     : Tuple			     =					merge(left.right	,A__						)
						return left
			else:
						UpperCAmelCase_     : int			     =					merge(A__	,right.left						)
						return right

def 				snake_case      (   A__	,A__						):
			UpperCAmelCase_     : Tuple			     =					Node(A__						)
			UpperCAmelCase_     ,	UpperCAmelCase_     : int			     =					split(A__	,A__						)
			return merge(merge(A__	,A__						)	,A__						)

def 				snake_case      (   A__	,A__						):
			UpperCAmelCase_     ,	UpperCAmelCase_     : List[str]			     =					split(A__	,value - 1						)
			UpperCAmelCase_     ,	UpperCAmelCase_     : List[str]			     =					split(A__	,A__						)
			return merge(A__	,A__						)

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

def 				snake_case      (   A__	,A__						):
			for arg in args.split():
						if arg[0] == "+":
									UpperCAmelCase_     : Union[str, Any]			     =					insert(A__	,int(arg[1:]						)						)

						elif arg[0] == "-":
									UpperCAmelCase_     : Optional[Any]			     =					erase(A__	,int(arg[1:]						)						)

						else:
									print("Unknown command"						)

			return root

def 				snake_case      (   ):
			UpperCAmelCase_     : 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. "						)

			UpperCAmelCase_     : Union[str, Any]			     =					input()
			while args != "q":
						UpperCAmelCase_     : Union[str, Any]			     =					interact_treap(A__	,A__						)
						print(A__						)
						UpperCAmelCase_     : List[Any]			     =					input()

			print("good by!"						)


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			main()











"""simple docstring"""
def 				snake_case      (   A__	,A__						):
			# "extended trapezoidal rule"
			# int(f) = dx/2 * (f1 + 2f2 + ... + fn)
			UpperCAmelCase_     : Dict			     =					(boundary[1] - boundary[0]) / steps
			UpperCAmelCase_     : Optional[int]			     =					boundary[0]
			UpperCAmelCase_     : str			     =					boundary[1]
			UpperCAmelCase_     : Tuple			     =					make_points(A__	,A__	,A__						)
			UpperCAmelCase_     : List[str]			     =					0.0
			y += (h / 2.0) * f(A__						)
			for i in x_i:
						# print(i)
						y += h * f(A__						)
			y += (h / 2.0) * f(A__						)
			return y

def 				snake_case      (   A__	,A__	,A__						):
			UpperCAmelCase_     : Union[str, Any]			     =					a + h
			while x < (b - h):
						yield x
						UpperCAmelCase_     : Optional[Any]			     =					x + h

def 				snake_case      (   A__						):  # enter your function here
			UpperCAmelCase_     : Dict			     =					(x - 0) * (x - 0)
			return y

def 				snake_case      (   ):
			UpperCAmelCase_     : Dict			     =					0.0  # Lower bound of integration
			UpperCAmelCase_     : Optional[int]			     =					1.0  # Upper bound of integration
			UpperCAmelCase_     : Dict			     =					10.0  # define number of steps or resolution
			UpperCAmelCase_     : List[Any]			     =					[a, b]  # define boundary of integration
			UpperCAmelCase_     : Union[str, Any]			     =					method_a(A__	,A__						)
			print(F"""y = {y}"""						)


if __name__ == "__main__":
			main()



