
import functools
from math import sqrt
from time import time




def 						A_			(				snake_case				: str			,   *snake_case				: Union[str, Any]			,   **snake_case				: int			)   ->     List[Any]:



						'''simple docstring'''

						__UpperCamelCase  	=      time()
						__UpperCamelCase  	=      func(*snake_case			,   **snake_case			)
						__UpperCamelCase  	=      time()
						if int(end - start			) > 0:
												print(f"{func.__name__} runtime: {(end - start):0.4f} s"			)
						else:
												print(f"{func.__name__} runtime: {(end - start) * 1000:0.4f} ms"			)
						return output




def 						A_			(				snake_case				: int			)   ->     list[int]:



						'''simple docstring'''

						if n < 0:
												raise Exception('''n is negative'''			)
						if n == 0:
												return [0]
						__UpperCamelCase  	=      [0, 1]
						for _ in range(n - 1			):
												fib.append(fib[-1] + fib[-2]			)
						return fib




def 						A_			(				snake_case				: int			)   ->     list[int]:



						'''simple docstring'''

						def fib_recursive_term(snake_case				: int			) -> int:
												if i < 0:
																		raise Exception('''n is negative'''			)
												if i < 2:
																		return i
												return fib_recursive_term(i - 1			) + fib_recursive_term(i - 2			)

						if n < 0:
												raise Exception('''n is negative'''			)
						return [fib_recursive_term(snake_case			) for i in range(n + 1			)]




def 						A_			(				snake_case				: int			)   ->     list[int]:



						'''simple docstring'''

						@functools.cache
						def fib_recursive_term(snake_case				: int			) -> int:
												if i < 0:
																		raise Exception('''n is negative'''			)
												if i < 2:
																		return i
												return fib_recursive_term(i - 1			) + fib_recursive_term(i - 2			)

						if n < 0:
												raise Exception('''n is negative'''			)
						return [fib_recursive_term(snake_case			) for i in range(n + 1			)]




def 						A_			(				snake_case				: int			)   ->     list[int]:



						'''simple docstring'''

						if n < 0:
												raise Exception('''n is negative'''			)
						# Cache must be outside recursuive function
						# other it will reset every time it calls itself.
						__UpperCamelCase  	=      {0: 0, 1: 1, 2: 1}  # Prefilled cache

						def rec_fn_memoized(snake_case				: int			) -> int:
												if num in cache:
																		return cache[num]

												__UpperCamelCase  	=      rec_fn_memoized(num - 1			) + rec_fn_memoized(num - 2			)
												__UpperCamelCase  	=      value
												return value

						return [rec_fn_memoized(snake_case			) for i in range(n + 1			)]




def 						A_			(				snake_case				: int			)   ->     list[int]:



						'''simple docstring'''

						if n < 0:
												raise Exception('''n is negative'''			)
						if n >= 1475:
												raise Exception('''n is too large'''			)
						__UpperCamelCase  	=      sqrt(5			)
						__UpperCamelCase  	=      (1 + sqrt_a) / 2
						return [round(phi**i / sqrt_a			) for i in range(n + 1			)]


if __name__ == "__main__":
				lowercase__				:				Optional[Any]						=      3_0
				time_func(fib_iterative, num)
				time_func(fib_recursive, num)  # Around 3s runtime
				time_func(fib_recursive_cached, num)  # Around 0ms runtime
				time_func(fib_memoization, num)
				time_func(fib_binet, num)












from __future__ import annotations




def       _a   (							lowerCamelCase__				)  ->      list[int]:
 lowerCamelCase_							:    Any							=     [0] * max_one

 for i in range(0 ,			lowerCamelCase__				):
  lowerCamelCase_							:    Union[str, Any]							=     i

 for i in range(2 ,			lowerCamelCase__				):
  if totients[i] == i:
   for j in range(lowerCamelCase__ ,			lowerCamelCase__ ,			lowerCamelCase__				):
    totients[j] -= totients[j] // i

 return totients




def       _a   (							lowerCamelCase__ ,			lowerCamelCase__				)  ->      bool:
 return sorted(str(lowerCamelCase__				)				) == sorted(str(lowerCamelCase__				)				)




def       _a   (							lowerCamelCase__ = 10_00_00_00				)  ->      int:
 lowerCamelCase_							:    Tuple							=     1  # i
 lowerCamelCase_							:    Union[str, Any]							=     0  # φ(i)
 lowerCamelCase_							:    List[Any]							=     get_totients(max_n + 1				)

 for i in range(2 ,			max_n + 1				):
  lowerCamelCase_							:    Tuple							=     totients[i]

  if i * min_denominator < min_numerator * t and has_same_digits(lowerCamelCase__ ,			lowerCamelCase__				):
   lowerCamelCase_							:    str							=     i
   lowerCamelCase_							:    List[Any]							=     t

 return min_numerator


if __name__ == "__main__":
       print(f'''{solution() = }''')
