



from math import sqrt






def       lowerCAmelCase__						(       lowerCamelCase_						:						int):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Tuple				=	True

				# 0 and 1 are none primes.
				if number <= 1:
								lowerCAmelCase__	:							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:
												lowerCAmelCase__	:							Union[str, Any]				=	False
												break

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

				return status






def       lowerCAmelCase__						(       lowerCamelCase_						:						Tuple):





				'''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
				lowerCAmelCase__	:							List[str]				=	list(range(2			,n + 1))

				lowerCAmelCase__	:							List[str]				=	[]  # 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):
																lowerCAmelCase__	:							List[Any]				=	0

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

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

				return ans






def       lowerCAmelCase__						(       lowerCamelCase_						:						str):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Optional[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       lowerCAmelCase__						(       lowerCamelCase_						:						Optional[int]):





				'''simple docstring'''

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

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

				# potential prime number factors.

				lowerCAmelCase__	:							Tuple				=	2

				lowerCAmelCase__	:							int				=	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       lowerCAmelCase__						(       lowerCamelCase_						:						List[Any]):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Dict				=	0

				# prime factorization of 'number'
				lowerCAmelCase__	:							Dict				=	prime_factorization(lowerCAmelCase__)

				lowerCAmelCase__	:							Optional[int]				=	max(lowerCAmelCase__)

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

				return ans






def       lowerCAmelCase__						(       lowerCamelCase_						:						int):





				'''simple docstring'''

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

				lowerCAmelCase__	:							List[str]				=	0

				# prime factorization of 'number'
				lowerCAmelCase__	:							Tuple				=	prime_factorization(lowerCAmelCase__)

				lowerCAmelCase__	:							Union[str, Any]				=	min(lowerCAmelCase__)

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

				return ans






def       lowerCAmelCase__						(       lowerCamelCase_						:						List[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       lowerCAmelCase__						(       lowerCamelCase_						:						Tuple):





				'''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       lowerCAmelCase__						(       lowerCamelCase_						:						Optional[int]):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Dict				=	[]  # this list will returned

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

				# run variable for while-loops.
				lowerCAmelCase__	:							Tuple				=	0
				lowerCAmelCase__	:							str				=	None

				# exit variable. for break up the loops
				lowerCAmelCase__	:							List[Any]				=	True

				while i < len_pn and loop:
								lowerCAmelCase__	:							Optional[int]				=	i + 1

								while j < len_pn and loop:
												if prime_numbers[i] + prime_numbers[j] == number:
																lowerCAmelCase__	:							List[str]				=	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       lowerCAmelCase__						(       lowerCamelCase_						:						List[Any]			,lowerCamelCase_						:						Optional[int]):





				'''simple docstring'''

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

				lowerCAmelCase__	:							int				=	0

				while numbera != 0:
								lowerCAmelCase__	:							Tuple				=	numbera % numbera
								lowerCAmelCase__	:							Optional[Any]				=	numbera
								lowerCAmelCase__	:							str				=	rest

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

				return numbera






def       lowerCAmelCase__						(       lowerCamelCase_						:						List[Any]			,lowerCamelCase_						:						List[Any]):





				'''simple docstring'''

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

				lowerCAmelCase__	:							List[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'
								lowerCAmelCase__	:							Dict				=	prime_factorization(lowerCAmelCase__)
								lowerCAmelCase__	:							List[str]				=	prime_factorization(lowerCAmelCase__)

				elif numbera == 1 or numbera == 1:
								lowerCAmelCase__	:							Dict				=	[]
								lowerCAmelCase__	:							List[Any]				=	[]
								lowerCAmelCase__	:							List[str]				=	max(lowerCAmelCase__			,lowerCAmelCase__)

				lowerCAmelCase__	:							str				=	0
				lowerCAmelCase__	:							Dict				=	0

				lowerCAmelCase__	:							List[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:
																lowerCAmelCase__	:							Tuple				=	prime_fac_a.count(lowerCAmelCase__)
																lowerCAmelCase__	:							Optional[Any]				=	prime_fac_a.count(lowerCAmelCase__)

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

												else:
																lowerCAmelCase__	:							Tuple				=	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:
												lowerCAmelCase__	:							Tuple				=	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       lowerCAmelCase__						(       lowerCamelCase_						:						Tuple):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Tuple				=	0
				lowerCAmelCase__	:							List[str]				=	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       lowerCAmelCase__						(       lowerCamelCase_						:						List[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'"

				lowerCAmelCase__	:							Optional[Any]				=	p_number_a + 1  # jump to the next number

				lowerCAmelCase__	:							Tuple				=	[]  # 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       lowerCAmelCase__						(       lowerCamelCase_						:						List[Any]):





				'''simple docstring'''

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

				lowerCAmelCase__	:							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       lowerCAmelCase__						(       lowerCamelCase_						:						Optional[int]):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Union[str, Any]				=	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       lowerCAmelCase__						(       lowerCamelCase_						:						Any			,lowerCamelCase_						:						Union[str, 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.
				lowerCAmelCase__	:							Any				=	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       lowerCAmelCase__						(       lowerCamelCase_						:						Dict):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Any				=	1  # this will be return.

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

				return ans






def       lowerCAmelCase__						(       lowerCamelCase_						:						List[str]):





				'''simple docstring'''

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

				lowerCAmelCase__	:							Dict				=	0
				lowerCAmelCase__	:							List[Any]				=	1
				lowerCAmelCase__	:							Any				=	1  # this will be return

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

				return ans






from bisect import bisect
from itertools import accumulate






def  lowerCamelCase_	(      lowerCAmelCase__	:	int			,	lowerCAmelCase__	:	str			,	lowerCAmelCase__	:	List[str]			,	lowerCAmelCase__	:	List[Any]				)       ->							Union[str, Any]:

							'''simple docstring'''






							A      						=    sorted(zip(lowerCAmelCase__			,	lowerCAmelCase__				)			,	key=lambda lowerCAmelCase__				: x[0] / x[1]			,	reverse=lowerCAmelCase__				)
							A						,   A      						=    [i[0] for i in r], [i[1] for i in r]
							A      						=    list(accumulate(lowerCAmelCase__				)				)
							A      						=    bisect(lowerCAmelCase__			,	lowerCAmelCase__				)
							return (
							    0
							    if k == 0
							    else sum(vl[:k]				) + (w - acc[k - 1]) * (vl[k]) / (wt[k])
							    if k != n
							    else sum(vl[:k]				)
							)


if __name__ == "__main__":
				import doctest

				doctest.testmod()