






from functools import lru_cache







@lru_cache
def   snake_case		(				UpperCAmelCase						:							int ):
				if not isinstance(UpperCAmelCase,    UpperCAmelCase ):
								raise TypeError('k must be an integer.' )
				if k < 0:
								raise ValueError('k integer must be greater or equal to zero.' )
				if k == 0:
								return 0
				elif k == 1:
								return 1
				else:
								return fibonacci(k - 1 ) + fibonacci(k - 2 )







def   snake_case		(				UpperCAmelCase						:							list,    UpperCAmelCase						:							int ):
				A									=   len(UpperCAmelCase )
				# Find m such that F_m >= n where F_i is the i_th fibonacci number.
				A									=   0
				while True:
								if fibonacci(UpperCAmelCase ) >= len_list:
												A									=   i
												break
								i += 1
				A									=   0
				while fibb_k > 0:
								A									=   min(
								    offset + fibonacci(fibb_k - 1 ),    len_list - 1 )  # Prevent out of range
								A									=   arr[index_k]
								if item_k_1 == val:
												return index_k
								elif val < item_k_1:
												fibb_k -= 1
								elif val > item_k_1:
												offset += fibonacci(fibb_k - 1 )
												fibb_k -= 2
				else:
								return -1


if __name__ == "__main__":
					import doctest

					doctest.testmod()













from __future__ import annotations







def   snake_case		(				UpperCAmelCase						:							int,    UpperCAmelCase						:							list[int],    UpperCAmelCase						:							list[int],    UpperCAmelCase						:							int ):

				# Base Case
				if counter == 0 or capacity == 0:
								return 0

				# If weight of the nth item is more than Knapsack of capacity,
				#   then this item cannot be included in the optimal solution,
				# else return the maximum of two cases:
				#   (1) nth item included
				#   (2) not included
				if weights[counter - 1] > capacity:
								return knapsack(UpperCAmelCase,    UpperCAmelCase,    UpperCAmelCase,    counter - 1 )
				else:
								A									=   capacity - weights[counter - 1]
								A									=   values[counter - 1] + knapsack(
								    UpperCAmelCase,    UpperCAmelCase,    UpperCAmelCase,    counter - 1 )
								A									=   knapsack(UpperCAmelCase,    UpperCAmelCase,    UpperCAmelCase,    counter - 1 )
								return max(UpperCAmelCase,    UpperCAmelCase )


if __name__ == "__main__":
					import doctest

					doctest.testmod()





