





def lowerCAmelCase__(__snake_case	,__snake_case							)    ->				Tuple:







	'''simple docstring'''







	lowerCamelCase__        =		[]
	lowerCamelCase__        =		len(_A							)
	for i in range(_A							):
		lowerCamelCase__        =		[]
		for j in range(_A							):
			lowerCamelCase__        =		0
			for k in range(_A							):
				lowerCamelCase__        =		val + matrix_a[i][k] * matrix_b[k][j]
			list_a.append(_A							)
		matrix_c.append(_A							)
	return matrix_c





def lowerCAmelCase__(__snake_case							)    ->				int:







	'''simple docstring'''







	return [[int(row == column							) for column in range(_A							)] for row in range(_A							)]





def lowerCAmelCase__(__snake_case							)    ->				Union[str, Any]:







	'''simple docstring'''







	if n <= 1:
		return n
	lowerCamelCase__        =		identity(2							)
	lowerCamelCase__        =		[[1, 1], [1, 0]]
	lowerCamelCase__        =		n - 1
	while n > 0:
		if n % 2 == 1:
			lowerCamelCase__        =		multiply(_A	,_A							)
		lowerCamelCase__        =		multiply(_A	,_A							)
		lowerCamelCase__        =		int(n / 2							)
	return res_matrix[0][0]





def lowerCAmelCase__(__snake_case							)    ->				List[str]:







	'''simple docstring'''







	if n <= 1:
		return n
	lowerCamelCase__        =		0
	lowerCamelCase__        =		1
	for _ in range(2	,n + 1							):
		lowerCamelCase__    ,     lowerCamelCase__        =		fiba, fiba + fiba
	return fiba





def lowerCAmelCase__()    ->				str:







	'''simple docstring'''







	for ordinal in "0th 1st 2nd 3rd 10th 100th 1000th".split():
		lowerCamelCase__        =		int(''''''.join(c for c in ordinal if c in '''0123456789'''							)							)  # 1000th --> 1000
		print(
		    F'{ordinal} fibonacci number using matrix exponentiation is '
		    F'{nth_fibonacci_matrix(_A							)} and using bruteforce is '
		    F'{nth_fibonacci_bruteforce(_A							)}\n'							)
	# from timeit import timeit
	# print(timeit("nth_fibonacci_matrix(1000000)",
	#              "from main import nth_fibonacci_matrix", number=5))
	# print(timeit("nth_fibonacci_bruteforce(1000000)",
	#              "from main import nth_fibonacci_bruteforce", number=5))
	# 2.3342058970001744
	# 57.256506615000035


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			main()










from __future__ import annotations

import math
import random





def lowerCAmelCase__(__snake_case	,__snake_case	,__snake_case							)    ->				list[int]:







	'''simple docstring'''







	lowerCamelCase__        =		d * e - 1
	lowerCamelCase__        =		0
	lowerCamelCase__        =		0
	while p == 0:
		lowerCamelCase__        =		random.randint(2	,n - 1							)
		lowerCamelCase__        =		k
		while True:
			if t % 2 == 0:
				lowerCamelCase__        =		t // 2
				lowerCamelCase__        =		(g**t) % n
				lowerCamelCase__        =		math.gcd(x - 1	,__snake_case							)
				if x > 1 and y > 1:
					lowerCamelCase__        =		y
					lowerCamelCase__        =		n // y
					break  # find the correct factors
			else:
				break  # t is not divisible by 2, break and choose another g
	return sorted([p, q]							)


if __name__ == "__main__":
			import doctest

			doctest.testmod()



