





import numpy as np




def 		lowerCAmelCase_    (  __UpperCAmelCase:			Tuple		,   __UpperCAmelCase:			Optional[Any]		,   __UpperCAmelCase:			str		,   __UpperCAmelCase:			List[Any]		,   __UpperCAmelCase:			List[str]      )      ->	Optional[int]:
    UpperCamelCase__							:				List[str]          =      int(np.ceil((x_end - xa) / h      )      )
    UpperCamelCase__							:				Any          =      np.zeros((n + 1,)      )
    UpperCamelCase__							:				Optional[int]          =      ya
    UpperCamelCase__							:				Dict          =      xa

    for k in range(lowercase__      ):
        UpperCamelCase__							:				Any          =      f(lowercase__		,   y[k]      )
        UpperCamelCase__							:				str          =      f(x + 0.5 * h		,   y[k] + 0.5 * h * ka      )
        UpperCamelCase__							:				Tuple          =      f(x + 0.5 * h		,   y[k] + 0.5 * h * ka      )
        UpperCamelCase__							:				Optional[Any]          =      f(x + h		,   y[k] + h * ka      )
        UpperCamelCase__							:				Tuple          =      y[k] + (1 / 6) * h * (ka + 2 * ka + 2 * ka + ka)
        x += h

    return y


if __name__ == "__main__":
  import doctest

  doctest.testmod()







"""simple docstring"""






import functools
from typing import Any







def      _snake_case			(				lowercase__							,    lowercase__		):

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

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

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

				for word in words:
								_lowerCamelCase     :			Dict    =    trie
								for c in word:
												if c not in trie_node:
																_lowerCamelCase     :			Any    =    {}

												_lowerCamelCase     :			str    =    trie_node[c]

								_lowerCamelCase     :			Optional[Any]    =    True

				_lowerCamelCase     :			Dict    =    len(lowercase__		)

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

								_lowerCamelCase     :			List[Any]    =    trie
								for i in range(lowercase__							,    lowercase__		):
												_lowerCamelCase     :			Any    =    trie_node.get(string[i]							,    lowercase__		)

												if trie_node is None:
																return False

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

								return False

				return is_breakable(0		)


if __name__ == "__main__":
			import doctest

			doctest.testmod()