
'''simple docstring'''







from math import sqrt
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			int							):



						"""simple docstring"""





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

						lowercase_     :					Optional[int]													=	True

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

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

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

						return status
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Tuple							):



						"""simple docstring"""





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

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

						lowercase_     :					int													=	[]  # this list will be returns.

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

    # filters actual prime numbers.
						lowercase_     :					int													=	[x for x in begin_list if x != 0]

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			List[Any]							):



						"""simple docstring"""





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

						lowercase_     :					List[str]													=	[]

						# 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(__SCREAMING_SNAKE_CASE							):
																		ans.append(__SCREAMING_SNAKE_CASE							)

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			List[Any]							):



						"""simple docstring"""





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

						lowercase_     :					List[str]													=	[]  # this list will be returns of the function.

						# potential prime number factors.

						lowercase_     :					Dict													=	2

						lowercase_     :					List[Any]													=	number

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

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

						else:
												ans.append(__SCREAMING_SNAKE_CASE							)

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Optional[int]							):



						"""simple docstring"""





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

						lowercase_     :					List[Any]													=	0

						# prime factorization of 'number'
						lowercase_     :					int													=	prime_factorization(__SCREAMING_SNAKE_CASE							)

						lowercase_     :					List[Any]													=	max(__SCREAMING_SNAKE_CASE							)

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Dict							):



						"""simple docstring"""





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

						lowercase_     :					Optional[int]													=	0

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

						lowercase_     :					Optional[int]													=	min(__SCREAMING_SNAKE_CASE							)

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			int							):



						"""simple docstring"""





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

						return number % 2 == 0
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Dict							):



						"""simple docstring"""





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

						return number % 2 != 0
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Any							):



						"""simple docstring"""





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

						lowercase_     :					List[Any]													=	[]  # this list will returned

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

						# run variable for while-loops.
						lowercase_     :					int													=	0
						lowercase_     :					Optional[int]													=	None

						# exit variable. for break up the loops
						lowercase_     :					int													=	True

						while i < len_pn and loop:
												lowercase_     :					Dict													=	i + 1

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

																		j += 1

												i += 1

						# precondition
						assert (
						    isinstance(__SCREAMING_SNAKE_CASE				,				__SCREAMING_SNAKE_CASE							)
						    and (len(__SCREAMING_SNAKE_CASE							) == 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_		(    __SCREAMING_SNAKE_CASE :			Optional[Any]				,				__SCREAMING_SNAKE_CASE :			str							):



						"""simple docstring"""





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

						lowercase_     :					Dict													=	0

						while numbera != 0:
												lowercase_     :					Optional[int]													=	numbera % numbera
												lowercase_     :					Tuple													=	numbera
												lowercase_     :					Union[str, Any]													=	rest

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

						return numbera
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Tuple				,				__SCREAMING_SNAKE_CASE :			int							):



						"""simple docstring"""





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

						lowercase_     :					Optional[int]													=	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'
												lowercase_     :					Any													=	prime_factorization(__SCREAMING_SNAKE_CASE							)
												lowercase_     :					str													=	prime_factorization(__SCREAMING_SNAKE_CASE							)

						elif numbera == 1 or numbera == 1:
												lowercase_     :					List[str]													=	[]
												lowercase_     :					List[Any]													=	[]
												lowercase_     :					Optional[int]													=	max(__SCREAMING_SNAKE_CASE				,				__SCREAMING_SNAKE_CASE							)

						lowercase_     :					Optional[int]													=	0
						lowercase_     :					Any													=	0

						lowercase_     :					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:
																								lowercase_     :					str													=	prime_fac_a.count(__SCREAMING_SNAKE_CASE							)
																								lowercase_     :					str													=	prime_fac_a.count(__SCREAMING_SNAKE_CASE							)

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

																		else:
																								lowercase_     :					Optional[Any]													=	prime_fac_a.count(__SCREAMING_SNAKE_CASE							)

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

																		done.append(__SCREAMING_SNAKE_CASE							)

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

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

																		done.append(__SCREAMING_SNAKE_CASE							)

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Union[str, Any]							):



						"""simple docstring"""





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

						lowercase_     :					Optional[int]													=	0
						lowercase_     :					Dict													=	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(__SCREAMING_SNAKE_CASE							):
																		ans += 1

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			List[Any]				,				__SCREAMING_SNAKE_CASE :			int							):



						"""simple docstring"""





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

						lowercase_     :					Optional[int]													=	p_number_a + 1  # jump to the next number

						lowercase_     :					Union[str, Any]													=	[]  # this list will be returns.

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

						while number < p_number_a:
												ans.append(__SCREAMING_SNAKE_CASE							)

												number += 1

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

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

						# 'ans' contains not 'pNumber1' and 'pNumber2' !
						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			Optional[int]							):



						"""simple docstring"""





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

						lowercase_     :					Tuple													=	[]  # will be returned.

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

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

						return ans
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			List[str]							):



						"""simple docstring"""





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

						lowercase_     :					Dict													=	get_divisors(__SCREAMING_SNAKE_CASE							)

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

						# summed all divisors up to 'number' (exclusive), hence [:-1]
						return sum(divisors[:-1]							) == number
def  snake_case_		(    __SCREAMING_SNAKE_CASE :			int				,				__SCREAMING_SNAKE_CASE :			Union[str, Any]							):



						"""simple docstring"""





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

						# build the greatest common divisor of numerator and denominator.
						lowercase_     :					List[str]													=	gcd(abs(__SCREAMING_SNAKE_CASE							)				,				abs(__SCREAMING_SNAKE_CASE							)							)

						# precondition
						assert (
						    isinstance(__SCREAMING_SNAKE_CASE				,				__SCREAMING_SNAKE_CASE							)
						    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_		(    __SCREAMING_SNAKE_CASE :			Dict							):



						"""simple docstring"""





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

						lowercase_     :					List[Any]													=	1  # this will be return.

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

						return ans




def  snake_case_		(    __SCREAMING_SNAKE_CASE :			List[str]							):



						"""simple docstring"""





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

						lowercase_     :					List[str]													=	0
						lowercase_     :					Dict													=	1
						lowercase_     :					Optional[Any]													=	1  # this will be return

						for _ in range(n - 1							):
												lowercase_     :					str													=	ans
												ans += fiba
												lowercase_     :					Any													=	tmp

						return ans





'''simple docstring'''







import logging
import os

from .state import PartialState






class 			lowerCAmelCase__       (  logging.LoggerAdapter					):






					@staticmethod
					def      _snake_case   (       __SCREAMING_SNAKE_CASE		):


											"""simple docstring"""





											lowercase_     :					Optional[Any]													=	PartialState()
											return not main_process_only or (main_process_only and state.is_main_process)






					def      _snake_case   (       self    ,							__SCREAMING_SNAKE_CASE    ,							__SCREAMING_SNAKE_CASE    ,							*__SCREAMING_SNAKE_CASE    ,							**__SCREAMING_SNAKE_CASE		):


											"""simple docstring"""





											if PartialState._shared_state == {}:
																	raise RuntimeError(
																	    '''You must initialize the accelerate state by calling either `PartialState()` or `Accelerator()` before using the logging utility.'''		)
											lowercase_     :					Tuple													=	kwargs.pop('''main_process_only'''    ,							__SCREAMING_SNAKE_CASE		)
											lowercase_     :					Optional[int]													=	kwargs.pop('''in_order'''    ,							__SCREAMING_SNAKE_CASE		)

											if self.isEnabledFor(__SCREAMING_SNAKE_CASE		):
																	if self._should_log(__SCREAMING_SNAKE_CASE		):
																							lowercase_						,			lowercase_     :					Optional[Any]													=	self.process(__SCREAMING_SNAKE_CASE    ,							__SCREAMING_SNAKE_CASE		)
																							self.logger.log(__SCREAMING_SNAKE_CASE    ,							__SCREAMING_SNAKE_CASE    ,							*__SCREAMING_SNAKE_CASE    ,							**__SCREAMING_SNAKE_CASE		)

																	elif in_order:
																							lowercase_     :					Optional[Any]													=	PartialState()
																							for i in range(state.num_processes		):
																													if i == state.process_index:
																																			lowercase_						,			lowercase_     :					Optional[int]													=	self.process(__SCREAMING_SNAKE_CASE    ,							__SCREAMING_SNAKE_CASE		)
																																			self.logger.log(__SCREAMING_SNAKE_CASE    ,							__SCREAMING_SNAKE_CASE    ,							*__SCREAMING_SNAKE_CASE    ,							**__SCREAMING_SNAKE_CASE		)
																													state.wait_for_everyone()







def  snake_case_		(    __SCREAMING_SNAKE_CASE :			str				,				__SCREAMING_SNAKE_CASE :			str = None							):



						"""simple docstring"""





						if log_level is None:
												lowercase_     :					Any													=	os.environ.get('''ACCELERATE_LOG_LEVEL'''				,				__SCREAMING_SNAKE_CASE							)
						lowercase_     :					List[str]													=	logging.getLogger(__SCREAMING_SNAKE_CASE							)
						if log_level is not None:
												logger.setLevel(log_level.upper()							)
												logger.root.setLevel(log_level.upper()							)
						return MultiProcessAdapter(__SCREAMING_SNAKE_CASE				,				{}							)



