
import functools
from typing import Any




def  __magic_name__ (   __lowerCAmelCase	:						str		,  __lowerCAmelCase	:						list[str]							)			->   bool:

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

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

				# Build trie
				__lowerCamelCase    					=	{}
				__lowerCamelCase    					=	'''WORD_KEEPER'''

				for word in words:
								__lowerCamelCase    					=	trie
								for c in word:
												if c not in trie_node:
																__lowerCamelCase    					=	{}

												__lowerCamelCase    					=	trie_node[c]

								__lowerCamelCase    					=	True

				__lowerCamelCase    					=	len(__lowerCAmelCase							)

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

								__lowerCamelCase    					=	trie
								for i in range(__lowerCAmelCase		,  __lowerCAmelCase							):
												__lowerCamelCase    					=	trie_node.get(string[i]		,  __lowerCAmelCase							)

												if trie_node is None:
																return False

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

								return False

				return is_breakable(0							)


if __name__ == "__main__":
					import doctest

					doctest.testmod()





from __future__ import annotations

from fractions import Fraction




def  __magic_name__ (   __lowerCAmelCase	:						int		,  __lowerCAmelCase	:						int							)			->   bool:
				return (
				    num != den and num % 10 == den // 10 and (num // 10) / (den % 10) == num / den
				)




def  __magic_name__ (   __lowerCAmelCase	:						int							)			->   list[str]:
				__lowerCamelCase    					=	[]
				__lowerCamelCase    					=	11
				__lowerCamelCase    					=	int('''1''' + '''0''' * digit_len							)
				for num in range(__lowerCAmelCase		,  __lowerCAmelCase							):
								while den <= 99:
												if (num != den) and (num % 10 == den // 10) and (den % 10 != 0):
																if is_digit_cancelling(__lowerCAmelCase		,  __lowerCAmelCase							):
																				solutions.append(f'''{num}/{den}'''							)
												den += 1
								num += 1
								__lowerCamelCase    					=	10
				return solutions




def  __magic_name__ (   __lowerCAmelCase	:						int = 2							)			->   int:
				__lowerCamelCase    					=	1.0
				for fraction in fraction_list(__lowerCAmelCase							):
								__lowerCamelCase    					=	Fraction(__lowerCAmelCase							)
								result *= frac.denominator / frac.numerator
				return int(__lowerCAmelCase							)


if __name__ == "__main__":
					print(solution())



