



from __future__ import annotations

from random import random






class    SCREAMING_SNAKE_CASE			:

							'''simple docstring'''





							def __init__(   self		:		str      ,				__lowercase		:		int | None = None	):

											'''simple docstring'''
											__a       =		value
											__a       =		random()
											__a       =		None
											__a       =		None





							def __repr__(   self		:		Any	):

											'''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		:		List[str]	):

											'''simple docstring'''
											__a       =		str(self.value	) + """ """
											__a       =		str(self.left or """"""	)
											__a       =		str(self.right or """"""	)
											return value + left + right



def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			Node | None				,     _SCREAMING_SNAKE_CASE       :			int    ):
				"""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:
												__a     ,					__a       =		split(root.left				,     _SCREAMING_SNAKE_CASE    )
												return left, root
								else:
												__a     ,					__a       =		split(root.right				,     _SCREAMING_SNAKE_CASE    )
												return root, right

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			Node | None				,     _SCREAMING_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:
								__a       =		merge(left.right				,     _SCREAMING_SNAKE_CASE    )
								return left
				else:
								__a       =		merge(_SCREAMING_SNAKE_CASE				,     right.left    )
								return right

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			Node | None				,     _SCREAMING_SNAKE_CASE       :			int    ):
				"""simple docstring"""







				__a       =		Node(_SCREAMING_SNAKE_CASE    )
				__a     ,					__a       =		split(_SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE    )
				return merge(merge(_SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE    )				,     _SCREAMING_SNAKE_CASE    )

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			Node | None				,     _SCREAMING_SNAKE_CASE       :			int    ):
				"""simple docstring"""







				__a     ,					__a       =		split(_SCREAMING_SNAKE_CASE				,     value - 1    )
				__a     ,					__a       =		split(_SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE    )
				return merge(_SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE    )

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			Node | None    ):
				"""simple docstring"""







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

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			Node | None				,     _SCREAMING_SNAKE_CASE       :			str    ):
				"""simple docstring"""







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

								elif arg[0] == "-":
												__a       =		erase(_SCREAMING_SNAKE_CASE				,     int(arg[1:]    )    )

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

				return root

def   lowerCAmelCase__						(							):
				"""simple docstring"""







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

				__a       =		input()
				while args != "q":
								__a       =		interact_treap(_SCREAMING_SNAKE_CASE				,     _SCREAMING_SNAKE_CASE    )
								print(_SCREAMING_SNAKE_CASE    )
								__a       =		input()

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


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						main()





from math import sqrt

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			int    ):
				"""simple docstring"""







				if 1 < number < 4:
								# 2 and 3 are primes
								return True
				elif number < 2 or number % 2 == 0 or number % 3 == 0:
								# Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
								return False

				# All primes number are in format of 6k +/- 1
				for i in range(5				,     int(sqrt(_SCREAMING_SNAKE_CASE    ) + 1    )				,     6    ):
								if number % i == 0 or number % (i + 2) == 0:
												return False
				return True

def   lowerCAmelCase__						(							_SCREAMING_SNAKE_CASE       :			int = 1_0001    ):
				"""simple docstring"""







				__a       =		0
				__a       =		1
				while count != nth and number < 3:
								number += 1
								if is_prime(_SCREAMING_SNAKE_CASE    ):
												count += 1
				while count != nth:
								number += 2
								if is_prime(_SCREAMING_SNAKE_CASE    ):
												count += 1
				return number


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