
def 	_SCREAMING_SNAKE_CASE					(		_lowerCamelCase :			int						,  _lowerCamelCase :			bool = False)					->    bool:






							'''simple docstring'''





							if n == 2:
														return True
							if not n % 2 or n < 2:
														return False
							if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
														return False
							if n > 3_317_044_064_679_887_385_961_981 and not allow_probable:
														raise ValueError(
														    "Warning: upper bound of deterministic test is exceeded. "
														    "Pass allow_probable=True to allow probabilistic test. "
														    "A return value of True indicates a probable prime.")
							# array bounds provided by analysis
							__UpperCamelCase	:   Optional[Any]			   =					[
							    2_047,
							    1_373_653,
							    25_326_001,
							    3_215_031_751,
							    2_152_302_898_747,
							    3_474_749_660_383,
							    341_550_071_728_321,
							    1,
							    3_825_123_056_546_413_051,
							    1,
							    1,
							    318_665_857_834_031_151_167_461,
							    3_317_044_064_679_887_385_961_981,
							]

							__UpperCamelCase	:   Dict			   =					[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
							for idx, _p in enumerate(_lowerCamelCase						,  1):
														if n < _p:
																					# then we have our last prime to check
																					__UpperCamelCase	:   List[Any]			   =					primes[:idx]
																					break
							__UpperCamelCase   ,						__UpperCamelCase	:   List[str]			   =					n - 1, 0
							# break up n -1 into a power of 2 (s) and
							# remaining odd component
							# essentially, solve for d * 2 ** s == n - 1
							while d % 2 == 0:
														d //= 2
														s += 1
							for prime in plist:
														__UpperCamelCase	:   str			   =					False
														for r in range(_lowerCamelCase):
																					__UpperCamelCase	:   Dict			   =					pow(_lowerCamelCase						,  d * 2**r						,  _lowerCamelCase)
																					# see article for analysis explanation for m
																					if (r == 0 and m == 1) or ((m + 1) % n == 0):
																												__UpperCamelCase	:   Union[str, Any]			   =					True
																												# this loop will not determine compositeness
																												break
														if pr:
																					continue
														# if pr is False, then the above loop never evaluated to true,
														# and the n MUST be composite
														return False
							return True




def 	_SCREAMING_SNAKE_CASE					(		)					->    None:






							'''simple docstring'''





							assert not miller_rabin(561)
							assert miller_rabin(563)
							# 2047

							assert not miller_rabin(838_201)
							assert miller_rabin(838_207)
							# 1_373_653

							assert not miller_rabin(17_316_001)
							assert miller_rabin(17_316_017)
							# 25_326_001

							assert not miller_rabin(3_078_386_641)
							assert miller_rabin(3_078_386_653)
							# 3_215_031_751

							assert not miller_rabin(1_713_045_574_801)
							assert miller_rabin(1_713_045_574_819)
							# 2_152_302_898_747

							assert not miller_rabin(2_779_799_728_307)
							assert miller_rabin(2_779_799_728_327)
							# 3_474_749_660_383

							assert not miller_rabin(113_850_023_909_441)
							assert miller_rabin(113_850_023_909_527)
							# 341_550_071_728_321

							assert not miller_rabin(1_275_041_018_848_804_351)
							assert miller_rabin(1_275_041_018_848_804_391)
							# 3_825_123_056_546_413_051

							assert not miller_rabin(79_666_464_458_507_787_791_867)
							assert miller_rabin(79_666_464_458_507_787_791_951)
							# 318_665_857_834_031_151_167_461

							assert not miller_rabin(552_840_677_446_647_897_660_333)
							assert miller_rabin(552_840_677_446_647_897_660_359)
							# 3_317_044_064_679_887_385_961_981
							# upper limit for probabilistic test


if __name__ == "__main__":
		test_miller_rabin()

class  lowerCamelCase__							:




	'''simple docstring'''




	def __init__(					self		:Union[str, Any]						)		-> None:
								__UpperCamelCase	:   dict[str, TrieNode]			   =					{}  # Mapping from char to TrieNode
								__UpperCamelCase	:   List[str]			   =					False







	def    _lowerCamelCase				(					self		:Any     ,   a		:list[str]						)		-> None:
								for word in words:
															self.insert(a						)







	def    _lowerCamelCase				(					self		:List[str]     ,   a		:str						)		-> None:
								__UpperCamelCase	:   Dict			   =					self
								for char in word:
															if char not in curr.nodes:
																						__UpperCamelCase	:   List[Any]			   =					TrieNode()
															__UpperCamelCase	:   List[Any]			   =					curr.nodes[char]
								__UpperCamelCase	:   Union[str, Any]			   =					True







	def    _lowerCamelCase				(					self		:Optional[int]     ,   a		:str						)		-> bool:
								__UpperCamelCase	:   Union[str, Any]			   =					self
								for char in word:
															if char not in curr.nodes:
																						return False
															__UpperCamelCase	:   Union[str, Any]			   =					curr.nodes[char]
								return curr.is_leaf







	def    _lowerCamelCase				(					self		:Any     ,   a		:str						)		-> None:

								def _delete(a		:TrieNode     ,   a		:str     ,   a		:int						) -> bool:
															if index == len(a						):
																						# If word does not exist
																						if not curr.is_leaf:
																													return False
																						__UpperCamelCase	:   str			   =					False
																						return len(curr.nodes						) == 0
															__UpperCamelCase	:   List[Any]			   =					word[index]
															__UpperCamelCase	:   Optional[int]			   =					curr.nodes.get(a						)
															# If char not in current trie node
															if not char_node:
																						return False
															# Flag to check if node can be deleted
															__UpperCamelCase	:   int			   =					_delete(a     ,   a     ,   index + 1						)
															if delete_curr:
																						del curr.nodes[char]
																						return len(curr.nodes						) == 0
															return delete_curr

								_delete(self     ,   a     ,   0						)




def 	_SCREAMING_SNAKE_CASE					(		_lowerCamelCase :			TrieNode						,  _lowerCamelCase :			str)					->    None:






							'''simple docstring'''





							if node.is_leaf:
														print(_lowerCamelCase						,  end=" ")

							for key, value in node.nodes.items():
														print_words(_lowerCamelCase						,  word + key)




def 	_SCREAMING_SNAKE_CASE					(		)					->    bool:






							'''simple docstring'''





							__UpperCamelCase	:   int			   =					"banana bananas bandana band apple all beast".split()
							__UpperCamelCase	:   Union[str, Any]			   =					TrieNode()
							root.insert_many(_lowerCamelCase)
							# print_words(root, "")
							assert all(root.find(_lowerCamelCase) for word in words)
							assert root.find("banana")
							assert not root.find("bandanas")
							assert not root.find("apps")
							assert root.find("apple")
							assert root.find("all")
							root.delete("all")
							assert not root.find("all")
							root.delete("banana")
							assert not root.find("banana")
							assert root.find("bananas")
							return True




def 	_SCREAMING_SNAKE_CASE					(		_lowerCamelCase :			str						,  _lowerCamelCase :			bool)					->    None:






							'''simple docstring'''





							print(str(_lowerCamelCase)						,  "works!" if passes else "doesn't work :(")




def 	_SCREAMING_SNAKE_CASE					(		)					->    None:






							'''simple docstring'''





							assert test_trie()




def 	_SCREAMING_SNAKE_CASE					(		)					->    None:






							'''simple docstring'''





							print_results("Testing trie functionality"						,  test_trie())


if __name__ == "__main__":
		main()