





from math import sqrt



def 						A_							(	_lowerCAmelCase	)		->					bool:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (
     number >= 0
 ), "'number' must been an int and positive"

 UpperCamelCase      :   List[str]			      =  True

 # 0 and 1 are none primes.
 if number <= 1:
  UpperCamelCase      :   Optional[int]			      =  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:
   UpperCamelCase      :   List[Any]			      =  False
   break

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

 return status



def 						A_							(	_lowerCAmelCase	)		->					Any:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (n > 2), "'N' must been an int and > 2"

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

 UpperCamelCase      :   Dict			      =  []  # 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):
    UpperCamelCase      :   str			      =  0

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

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

 return ans



def 						A_							(	_lowerCAmelCase	)		->					Union[str, Any]:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (n > 2), "'N' must been an int and > 2"

 UpperCamelCase      :   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(_lowerCAmelCase	):
   ans.append(_lowerCAmelCase	)

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

 return ans



def 						A_							(	_lowerCAmelCase	)		->					Union[str, Any]:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and number >= 0, "'number' must been an int and >= 0"

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

 # potential prime number factors.

 UpperCamelCase      :   List[Any]			      =  2

 UpperCamelCase      :   Dict			      =  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 						A_							(	_lowerCAmelCase	)		->					Optional[int]:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (
     number >= 0
 ), "'number' bust been an int and >= 0"

 UpperCamelCase      :   Tuple			      =  0

 # prime factorization of 'number'
 UpperCamelCase      :   int			      =  prime_factorization(_lowerCAmelCase	)

 UpperCamelCase      :   Tuple			      =  max(_lowerCAmelCase	)

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

 return ans



def 						A_							(	_lowerCAmelCase	)		->					int:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (
     number >= 0
 ), "'number' bust been an int and >= 0"

 UpperCamelCase      :   int			      =  0

 # prime factorization of 'number'
 UpperCamelCase      :   Optional[Any]			      =  prime_factorization(_lowerCAmelCase	)

 UpperCamelCase      :   Optional[int]			      =  min(_lowerCAmelCase	)

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

 return ans



def 						A_							(	_lowerCAmelCase	)		->					Optional[Any]:
 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 						A_							(	_lowerCAmelCase	)		->					str:
 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 						A_							(	_lowerCAmelCase	)		->					Any:
 assert (
     isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (number > 2) and is_even(_lowerCAmelCase	)
 ), "'number' must been an int, even and > 2"

 UpperCamelCase      :   List[str]			      =  []  # this list will returned

 # creates a list of prime numbers between 2 up to 'number'
 UpperCamelCase      :   int			      =  get_prime_numbers(_lowerCAmelCase	)
 UpperCamelCase      :   str			      =  len(_lowerCAmelCase	)

 # run variable for while-loops.
 UpperCamelCase      :   Any			      =  0
 UpperCamelCase      :   Tuple			      =  None

 # exit variable. for break up the loops
 UpperCamelCase      :   Tuple			      =  True

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

  while j < len_pn and loop:
   if prime_numbers[i] + prime_numbers[j] == number:
    UpperCamelCase      :   Tuple			      =  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 						A_							(	_lowerCAmelCase				,  _lowerCAmelCase	)		->					int:
 assert (
     isinstance(_lowerCAmelCase				,  _lowerCAmelCase	)
     and isinstance(_lowerCAmelCase				,  _lowerCAmelCase	)
     and (numbera >= 0)
     and (numbera >= 0)
 ), "'number1' and 'number2' must been positive integer."

 UpperCamelCase      :   Tuple			      =  0

 while numbera != 0:
  UpperCamelCase      :   List[Any]			      =  numbera % numbera
  UpperCamelCase      :   Optional[Any]			      =  numbera
  UpperCamelCase      :   List[str]			      =  rest

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

 return numbera



def 						A_							(	_lowerCAmelCase				,  _lowerCAmelCase	)		->					Union[str, Any]:
 assert (
     isinstance(_lowerCAmelCase				,  _lowerCAmelCase	)
     and isinstance(_lowerCAmelCase				,  _lowerCAmelCase	)
     and (numbera >= 1)
     and (numbera >= 1)
 ), "'number1' and 'number2' must been positive integer."

 UpperCamelCase      :   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'
  UpperCamelCase      :   Optional[Any]			      =  prime_factorization(_lowerCAmelCase	)
  UpperCamelCase      :   Optional[Any]			      =  prime_factorization(_lowerCAmelCase	)

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

 UpperCamelCase      :   int			      =  0
 UpperCamelCase      :   Union[str, Any]			      =  0

 UpperCamelCase      :   Dict			      =  []  # 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:
    UpperCamelCase      :   Optional[int]			      =  prime_fac_a.count(_lowerCAmelCase	)
    UpperCamelCase      :   str			      =  prime_fac_a.count(_lowerCAmelCase	)

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

   else:
    UpperCamelCase      :   List[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:
   UpperCamelCase      :   Any			      =  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 						A_							(	_lowerCAmelCase	)		->					List[Any]:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (n >= 0), "'number' must been a positive int"

 UpperCamelCase      :   int			      =  0
 UpperCamelCase      :   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 						A_							(	_lowerCAmelCase				,  _lowerCAmelCase	)		->					Dict:
 assert (
     is_prime(_lowerCAmelCase	) and is_prime(_lowerCAmelCase	) and (p_number_a < p_number_a)
 ), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

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

 UpperCamelCase      :   Union[str, 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 						A_							(	_lowerCAmelCase	)		->					int:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (n >= 1), "'n' must been int and >= 1"

 UpperCamelCase      :   Optional[Any]			      =  []  # 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 						A_							(	_lowerCAmelCase	)		->					Tuple:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (
     number > 1
 ), "'number' must been an int and >= 1"

 UpperCamelCase      :   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 						A_							(	_lowerCAmelCase				,  _lowerCAmelCase	)		->					Dict:
 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.
 UpperCamelCase      :   int			      =  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 						A_							(	_lowerCAmelCase	)		->					Union[str, Any]:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (n >= 0), "'n' must been a int and >= 0"

 UpperCamelCase      :   Tuple			      =  1  # this will be return.

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

 return ans



def 						A_							(	_lowerCAmelCase	)		->					Any:
 assert isinstance(_lowerCAmelCase				,  _lowerCAmelCase	) and (n >= 0), "'n' must been an int and >= 0"

 UpperCamelCase      :   Tuple			      =  0
 UpperCamelCase      :   Optional[Any]			      =  1
 UpperCamelCase      :   List[str]			      =  1  # this will be return

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

 return ans







from __future__ import annotations

import math



def 						A_							(	_lowerCAmelCase				,  _lowerCAmelCase	)		->					float:
 UpperCamelCase      :   Tuple			      =  u
 for i in range(1				,  _lowerCAmelCase	):
  UpperCamelCase      :   Any			      =  temp * (u - i)
 return temp



def 						A_							(	)		->					None:
 UpperCamelCase      :   Union[str, Any]			      =  int(input("enter the numbers of values: "	)	)
 UpperCamelCase      :   list[list[float]]			      =  []
 for _ in range(_lowerCAmelCase	):
  y.append([]	)
 for i in range(_lowerCAmelCase	):
  for j in range(_lowerCAmelCase	):
   y[i].append(_lowerCAmelCase	)
   UpperCamelCase      :   List[str]			      =  0

 print("enter the values of parameters in a list: "	)
 UpperCamelCase      :   Optional[Any]			      =  list(map(_lowerCAmelCase				,  input().split()	)	)

 print("enter the values of corresponding parameters: "	)
 for i in range(_lowerCAmelCase	):
  UpperCamelCase      :   Tuple			      =  float(input()	)

 UpperCamelCase      :   Any			      =  int(input("enter the value to interpolate: "	)	)
 UpperCamelCase      :   List[Any]			      =  (value - x[0]) / (x[1] - x[0])

 # for calculating forward difference table

 for i in range(1				,  _lowerCAmelCase	):
  for j in range(n - i	):
   UpperCamelCase      :   Optional[Any]			      =  y[j + 1][i - 1] - y[j][i - 1]

 UpperCamelCase      :   List[str]			      =  y[0][0]
 for i in range(1				,  _lowerCAmelCase	):
  summ += (ucal(_lowerCAmelCase				,  _lowerCAmelCase	) * y[0][i]) / math.factorial(_lowerCAmelCase	)

 print(F"""the value at {value} is {summ}"""	)


if __name__ == "__main__":
    main()
