





import functools
from typing import Any






def 					lowerCamelCase_						(    _UpperCamelCase			,    _UpperCamelCase			)		->						bool:






				"""simple docstring"""


				if not isinstance(_UpperCamelCase			,    _UpperCamelCase			) or len(_UpperCamelCase			) == 0:
								raise ValueError('''the string should be not empty string'''			)

				if not isinstance(_UpperCamelCase			,    _UpperCamelCase			) or not all(
				    isinstance(_UpperCamelCase			,    _UpperCamelCase			) and len(_UpperCamelCase			) > 0 for item in words			):
								raise ValueError('''the words should be a list of non-empty strings'''			)

				# Build trie
				snake_case_						:   dict[str, Any]           =  {}
				snake_case_						:   str           =  '''WORD_KEEPER'''

				for word in words:
								snake_case_						:   Optional[int]           =  trie
								for c in word:
												if c not in trie_node:
																snake_case_						:   str           =  {}

												snake_case_						:   Dict           =  trie_node[c]

								snake_case_						:   Optional[Any]           =  True

				snake_case_						:   Optional[int]           =  len(_UpperCamelCase			)

				# Dynamic programming method
				@functools.cache
				def is_breakable(_UpperCamelCase			) -> bool:
								if index == len_string:
												return True

								snake_case_						:   Optional[Any]           =  trie
								for i in range(_UpperCamelCase			,    _UpperCamelCase			):
												snake_case_						:   int           =  trie_node.get(string[i]			,    _UpperCamelCase			)

												if trie_node is None:
																return False

												if trie_node.get(_UpperCamelCase			,    _UpperCamelCase			) and is_breakable(i + 1			):
																return True

								return False

				return is_breakable(0			)


if __name__ == "__main__":
	import doctest

	doctest.testmod()







import random

from .binary_exp_mod import bin_exp_mod






def 					lowerCamelCase_						(    _UpperCamelCase			,    _UpperCamelCase=1_000			)		->						str:






				"""simple docstring"""


				if n < 2:
								return False

				if n % 2 == 0:
								return n == 2

				# this means n is odd
				snake_case_						:   int           =  n - 1
				snake_case_						:   Union[str, Any]           =  0
				while d % 2 == 0:
								d /= 2
								exp += 1

				# n - 1=d*(2**exp)
				snake_case_						:   List[Any]           =  0
				while count < prec:
								snake_case_						:   Tuple           =  random.randint(2			,    n - 1			)
								snake_case_						:   List[Any]           =  bin_exp_mod(_UpperCamelCase			,    _UpperCamelCase			,    _UpperCamelCase			)
								if b != 1:
												snake_case_						:   Any           =  True
												for _ in range(_UpperCamelCase			):
																if b == n - 1:
																				snake_case_						:   Tuple           =  False
																				break
																snake_case_						:   List[Any]           =  b * b
																b %= n
												if flag:
																return False
												count += 1
				return True


if __name__ == "__main__":
	lowerCAmelCase_    =    abs(int(input('''Enter bound : ''').strip()))
	print('''Here\'s the list of primes:''')
	print(''', '''.join(str(i) for i in range(n + 1) if is_prime_big(i)))
