
"""simple docstring"""

def 							A						(				snake_case      :Optional[Any]		)      ->	int:
 assert isinstance(__snake_case				,					__snake_case		), f'The input value of [n={number}] is not an integer'

 if number == 1:
  return 2
 elif number < 1:
  __UpperCamelCase				  =       f'The input value of [n={number}] has to be > 0'
  raise ValueError(__snake_case		)
 else:
  __UpperCamelCase				  =       sylvester(number - 1		)
  __UpperCamelCase				  =       num - 1
  __UpperCamelCase				  =       num
  return lower * upper + 1


if __name__ == "__main__":
      print(f'''The 8th number in Sylvester\'s sequence: {sylvester(8)}''')










from __future__ import annotations

from random import random






class 					__A    :

			'''simple docstring'''







			def __init__(    self       ,  __lowerCAmelCase = None							):



				'''simple docstring'''







				lowerCamelCase__								=		value
				lowerCamelCase__								=		random()
				lowerCamelCase__								=		None
				lowerCamelCase__								=		None





			def __repr__(    self							):



				'''simple docstring'''







				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							):



				'''simple docstring'''







				lowerCamelCase__								=		str(self.value							) + ''' '''
				lowerCamelCase__								=		str(self.left or ''''''							)
				lowerCamelCase__								=		str(self.right or ''''''							)
				return value + left + right





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      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:
			lowerCamelCase__				,	lowerCamelCase__								=		split(root.left       ,__snake_case							)
			return left, root
		else:
			lowerCamelCase__				,	lowerCamelCase__								=		split(root.right       ,__snake_case							)
			return root, right





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      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:
		lowerCamelCase__								=		merge(left.right       ,__snake_case							)
		return left
	else:
		lowerCamelCase__								=		merge(__snake_case       ,right.left							)
		return right





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      Node | None:







	'''simple docstring'''







	lowerCamelCase__								=		Node(__snake_case							)
	lowerCamelCase__				,	lowerCamelCase__								=		split(__snake_case       ,__snake_case							)
	return merge(merge(__snake_case       ,__snake_case							)       ,__snake_case							)





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      Node | None:







	'''simple docstring'''







	lowerCamelCase__				,	lowerCamelCase__								=		split(__snake_case       ,value - 1							)
	lowerCamelCase__				,	lowerCamelCase__								=		split(__snake_case       ,__snake_case							)
	return merge(__snake_case       ,__snake_case							)





def lowerCAmelCase__(__snake_case							)				->      None:







	'''simple docstring'''







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





def lowerCAmelCase__(__snake_case       ,__snake_case							)				->      Node | None:







	'''simple docstring'''







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

		elif arg[0] == "-":
			lowerCamelCase__								=		erase(__snake_case       ,int(arg[1:]							)							)

		else:
			print('''Unknown command'''							)

	return root





def lowerCAmelCase__()				->      None:







	'''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(__snake_case       ,__snake_case							)
		print(__snake_case							)
		lowerCamelCase__								=		input()

	print('''good by!'''							)


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			main()



