from math import sqrt




def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (
		    number >= 0
		), "'number' must been an int and positive"

		SCREAMING_SNAKE_CASE_  :		Optional[int]				      =    True

		# 0 and 1 are none primes.
		if number <= 1:
				SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    False

		for divisor in range(2   , int(round(sqrt(lowerCAmelCase	)	)	) + 1	):
				# if 'number' divisible by 'divisor' then sets 'status'
				# of false and break up the loop.
				if number % divisor == 0:
						SCREAMING_SNAKE_CASE_  :		int				      =    False
						break

    # precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'status' must been from type bool"

		return status




def       _snake_case	(   lowerCAmelCase							:   Dict	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (n > 2), "'N' must been an int and > 2"

		# beginList: contains all natural numbers from 2 up to N
		SCREAMING_SNAKE_CASE_  :		str				      =    list(range(2   , n + 1	)	)

		SCREAMING_SNAKE_CASE_  :		List[Any]				      =    []  # this list will be returns.

		# actual sieve of erathostenes
		for i in range(len(lowerCAmelCase	)	):
				for j in range(i + 1   , len(lowerCAmelCase	)	):
						if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
								SCREAMING_SNAKE_CASE_  :		List[str]				      =    0

    # filters actual prime numbers.
		SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    [x for x in begin_list if x != 0]

		# precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'ans' must been from type list"

		return ans




def       _snake_case	(   lowerCAmelCase							:   Dict	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (n > 2), "'N' must been an int and > 2"

		SCREAMING_SNAKE_CASE_  :		Any				      =    []

		# iterates over all numbers between 2 up to N+1
		# if a number is prime then appends to list 'ans'
		for number in range(2   , n + 1	):
				if is_prime(lowerCAmelCase	):
						ans.append(lowerCAmelCase	)

    # precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'ans' must been from type list"

		return ans




def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and number >= 0, "'number' must been an int and >= 0"

		SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    []  # this list will be returns of the function.

		# potential prime number factors.

		SCREAMING_SNAKE_CASE_  :		Optional[int]				      =    2

		SCREAMING_SNAKE_CASE_  :		List[Any]				      =    number

		if number == 0 or number == 1:
				ans.append(lowerCAmelCase	)

		# if 'number' not prime then builds the prime factorization of 'number'
		elif not is_prime(lowerCAmelCase	):
				while quotient != 1:
						if is_prime(lowerCAmelCase	) and (quotient % factor == 0):
								ans.append(lowerCAmelCase	)
								quotient /= factor
						else:
								factor += 1

		else:
				ans.append(lowerCAmelCase	)

		# precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'ans' must been from type list"

		return ans




def       _snake_case	(   lowerCAmelCase							:   str	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (
		    number >= 0
		), "'number' bust been an int and >= 0"

		SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    0

		# prime factorization of 'number'
		SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    prime_factorization(lowerCAmelCase	)

		SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    max(lowerCAmelCase	)

		# precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'ans' must been from type int"

		return ans




def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (
		    number >= 0
		), "'number' bust been an int and >= 0"

		SCREAMING_SNAKE_CASE_  :		Dict				      =    0

		# prime factorization of 'number'
		SCREAMING_SNAKE_CASE_  :		Any				      =    prime_factorization(lowerCAmelCase	)

		SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    min(lowerCAmelCase	)

		# precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'ans' must been from type int"

		return ans




def       _snake_case	(   lowerCAmelCase							:   Optional[Any]	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'number' must been an int"
		assert isinstance(number % 2 == 0   , lowerCAmelCase	), "compare bust been from type bool"

		return number % 2 == 0




def       _snake_case	(   lowerCAmelCase							:   Any	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	), "'number' must been an int"
		assert isinstance(number % 2 != 0   , lowerCAmelCase	), "compare bust been from type bool"

		return number % 2 != 0




def       _snake_case	(   lowerCAmelCase							:   List[str]	):
		"""simple docstring"""







		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	) and (number > 2) and is_even(lowerCAmelCase	)
		), "'number' must been an int, even and > 2"

		SCREAMING_SNAKE_CASE_  :		int				      =    []  # this list will returned

		# creates a list of prime numbers between 2 up to 'number'
		SCREAMING_SNAKE_CASE_  :		List[str]				      =    get_prime_numbers(lowerCAmelCase	)
		SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    len(lowerCAmelCase	)

		# run variable for while-loops.
		SCREAMING_SNAKE_CASE_  :		List[Any]				      =    0
		SCREAMING_SNAKE_CASE_  :		int				      =    None

		# exit variable. for break up the loops
		SCREAMING_SNAKE_CASE_  :		Optional[int]				      =    True

		while i < len_pn and loop:
				SCREAMING_SNAKE_CASE_  :		int				      =    i + 1

				while j < len_pn and loop:
						if prime_numbers[i] + prime_numbers[j] == number:
								SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    False
								ans.append(prime_numbers[i]	)
								ans.append(prime_numbers[j]	)

						j += 1

				i += 1

		# precondition
		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and (len(lowerCAmelCase	) == 2)
		    and (ans[0] + ans[1] == number)
		    and is_prime(ans[0]	)
		    and is_prime(ans[1]	)
		), "'ans' must contains two primes. And sum of elements must been eq 'number'"

		return ans




def       _snake_case	(   lowerCAmelCase							:   Optional[Any]   , lowerCAmelCase							:   Any	):
		"""simple docstring"""







		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and (numbera >= 0)
		    and (numbera >= 0)
		), "'number1' and 'number2' must been positive integer."

		SCREAMING_SNAKE_CASE_  :		str				      =    0

		while numbera != 0:
				SCREAMING_SNAKE_CASE_  :		str				      =    numbera % numbera
				SCREAMING_SNAKE_CASE_  :		Any				      =    numbera
				SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    rest

		# precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (
		    numbera >= 0
		), "'number' must been from type int and positive"

		return numbera




def       _snake_case	(   lowerCAmelCase							:   Any   , lowerCAmelCase							:   Tuple	):
		"""simple docstring"""







		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and (numbera >= 1)
		    and (numbera >= 1)
		), "'number1' and 'number2' must been positive integer."

		SCREAMING_SNAKE_CASE_  :		Any				      =    1  # actual answer that will be return.

		# for kgV (x,1)
		if numbera > 1 and numbera > 1:
				# builds the prime factorization of 'number1' and 'number2'
				SCREAMING_SNAKE_CASE_  :		str				      =    prime_factorization(lowerCAmelCase	)
				SCREAMING_SNAKE_CASE_  :		List[str]				      =    prime_factorization(lowerCAmelCase	)

		elif numbera == 1 or numbera == 1:
				SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    []
				SCREAMING_SNAKE_CASE_  :		Any				      =    []
				SCREAMING_SNAKE_CASE_  :		int				      =    max(lowerCAmelCase   , lowerCAmelCase	)

		SCREAMING_SNAKE_CASE_  :		Any				      =    0
		SCREAMING_SNAKE_CASE_  :		str				      =    0

		SCREAMING_SNAKE_CASE_  :		Any				      =    []  # captured numbers int both 'primeFac1' and 'primeFac2'

		# iterates through primeFac1
		for n in prime_fac_a:
				if n not in done:
						if n in prime_fac_a:
								SCREAMING_SNAKE_CASE_  :		List[str]				      =    prime_fac_a.count(lowerCAmelCase	)
								SCREAMING_SNAKE_CASE_  :		int				      =    prime_fac_a.count(lowerCAmelCase	)

								for _ in range(max(lowerCAmelCase   , lowerCAmelCase	)	):
										ans *= n

						else:
								SCREAMING_SNAKE_CASE_  :		Union[str, Any]				      =    prime_fac_a.count(lowerCAmelCase	)

								for _ in range(lowerCAmelCase	):
										ans *= n

						done.append(lowerCAmelCase	)

    # iterates through primeFac2
		for n in prime_fac_a:
				if n not in done:
						SCREAMING_SNAKE_CASE_  :		List[str]				      =    prime_fac_a.count(lowerCAmelCase	)

						for _ in range(lowerCAmelCase	):
								ans *= n

						done.append(lowerCAmelCase	)

    # precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (
		    ans >= 0
		), "'ans' must been from type int and positive"

		return ans




def       _snake_case	(   lowerCAmelCase							:   Optional[Any]	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (n >= 0), "'number' must been a positive int"

		SCREAMING_SNAKE_CASE_  :		str				      =    0
		SCREAMING_SNAKE_CASE_  :		Optional[int]				      =    2  # this variable holds the answer

		while index < n:
				index += 1

				ans += 1  # counts to the next number

				# if ans not prime then
				# runs to the next prime number.
				while not is_prime(lowerCAmelCase	):
						ans += 1

    # precondition
		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and is_prime(
		    lowerCAmelCase	), "'ans' must been a prime number and from type int"

		return ans




def       _snake_case	(   lowerCAmelCase							:   Union[str, Any]   , lowerCAmelCase							:   str	):
		"""simple docstring"""







		assert (
		    is_prime(lowerCAmelCase	) and is_prime(lowerCAmelCase	) and (p_number_a < p_number_a)
		), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

		SCREAMING_SNAKE_CASE_  :		List[Any]				      =    p_number_a + 1  # jump to the next number

		SCREAMING_SNAKE_CASE_  :		List[Any]				      =    []  # this list will be returns.

		# if number is not prime then
		# fetch the next prime number.
		while not is_prime(lowerCAmelCase	):
				number += 1

		while number < p_number_a:
				ans.append(lowerCAmelCase	)

				number += 1

				# fetch the next prime number.
				while not is_prime(lowerCAmelCase	):
						number += 1

    # precondition
		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and ans[0] != p_number_a
		    and ans[len(lowerCAmelCase	) - 1] != p_number_a
		), "'ans' must been a list without the arguments"

		# 'ans' contains not 'pNumber1' and 'pNumber2' !
		return ans




def       _snake_case	(   lowerCAmelCase							:   Optional[Any]	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (n >= 1), "'n' must been int and >= 1"

		SCREAMING_SNAKE_CASE_  :		List[str]				      =    []  # will be returned.

		for divisor in range(1   , n + 1	):
				if n % divisor == 0:
						ans.append(lowerCAmelCase	)

    # precondition
		assert ans[0] == 1 and ans[len(lowerCAmelCase	) - 1] == n, "Error in function getDivisiors(...)"

		return ans




def       _snake_case	(   lowerCAmelCase							:   Dict	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (
		    number > 1
		), "'number' must been an int and >= 1"

		SCREAMING_SNAKE_CASE_  :		int				      =    get_divisors(lowerCAmelCase	)

		# precondition
		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and (divisors[0] == 1)
		    and (divisors[len(lowerCAmelCase	) - 1] == number)
		), "Error in help-function getDivisiors(...)"

		# summed all divisors up to 'number' (exclusive), hence [:-1]
		return sum(divisors[:-1]	) == number




def       _snake_case	(   lowerCAmelCase							:   List[Any]   , lowerCAmelCase							:   Any	):
		"""simple docstring"""







		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and (denominator != 0)
		), "The arguments must been from type int and 'denominator' != 0"

		# build the greatest common divisor of numerator and denominator.
		SCREAMING_SNAKE_CASE_  :		Dict				      =    gcd(abs(lowerCAmelCase	)   , abs(lowerCAmelCase	)	)

		# precondition
		assert (
		    isinstance(lowerCAmelCase   , lowerCAmelCase	)
		    and (numerator % gcd_of_fraction == 0)
		    and (denominator % gcd_of_fraction == 0)
		), "Error in function gcd(...,...)"

		return (numerator // gcd_of_fraction, denominator // gcd_of_fraction)




def       _snake_case	(   lowerCAmelCase							:   Tuple	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (n >= 0), "'n' must been a int and >= 0"

		SCREAMING_SNAKE_CASE_  :		str				      =    1  # this will be return.

		for factor in range(1   , n + 1	):
				ans *= factor

		return ans







def       _snake_case	(   lowerCAmelCase							:   int	):
		"""simple docstring"""







		assert isinstance(lowerCAmelCase   , lowerCAmelCase	) and (n >= 0), "'n' must been an int and >= 0"

		SCREAMING_SNAKE_CASE_  :		Tuple				      =    0
		SCREAMING_SNAKE_CASE_  :		Tuple				      =    1
		SCREAMING_SNAKE_CASE_  :		Any				      =    1  # this will be return

		for _ in range(n - 1	):
				SCREAMING_SNAKE_CASE_  :		str				      =    ans
				ans += fiba
				SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    tmp

		return ans


import PIL.Image
import PIL.ImageOps
from packaging import version
from PIL import Image


if version.parse(version.parse(PIL.__version__).base_version) >= version.parse('9.1.0'):
		lowercase  :					int													=       {
		    'linear': PIL.Image.Resampling.BILINEAR,
		    'bilinear': PIL.Image.Resampling.BILINEAR,
		    'bicubic': PIL.Image.Resampling.BICUBIC,
		    'lanczos': PIL.Image.Resampling.LANCZOS,
		    'nearest': PIL.Image.Resampling.NEAREST,
		}
else:
		lowercase  :					Any													=       {
		    'linear': PIL.Image.LINEAR,
		    'bilinear': PIL.Image.BILINEAR,
		    'bicubic': PIL.Image.BICUBIC,
		    'lanczos': PIL.Image.LANCZOS,
		    'nearest': PIL.Image.NEAREST,
		}




def 	_SCREAMING_SNAKE_CASE					(		_lowerCamelCase :			int)					->    Any:






							'''simple docstring'''





							__UpperCamelCase	:   Union[str, Any]			   =					(images / 2 + 0.5).clamp(0						,  1)
							__UpperCamelCase	:   int			   =					images.cpu().permute(0						,  2						,  3						,  1).float().numpy()
							__UpperCamelCase	:   Dict			   =					numpy_to_pil(_lowerCamelCase)
							return images




def 	_SCREAMING_SNAKE_CASE					(		_lowerCamelCase :			Optional[Any])					->    Optional[Any]:






							'''simple docstring'''





							if images.ndim == 3:
														__UpperCamelCase	:   Optional[int]			   =					images[None, ...]
							__UpperCamelCase	:   Dict			   =					(images * 255).round().astype("uint8")
							if images.shape[-1] == 1:
														# special case for grayscale (single channel) images
														__UpperCamelCase	:   int			   =					[Image.fromarray(image.squeeze()						,  mode="L") for image in images]
							else:
														__UpperCamelCase	:   Union[str, Any]			   =					[Image.fromarray(_lowerCamelCase) for image in images]

							return pil_images