






"""simple docstring"""





from math import sqrt



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

						__magic_name__:	List[str]							=       True

						# 0 and 1 are none primes.
						if number <= 1:
												__magic_name__:	List[str]							=       False

						for divisor in range(2      ,   int(round(sqrt(__UpperCAmelCase  )  )  ) + 1  ):
												# if 'number' divisible by 'divisor' then sets 'status'
												# of false and break up the loop.
												if number % divisor == 0:
																		__magic_name__:	Union[str, Any]							=       False
																		break

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

						return status



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

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

						__magic_name__:	List[str]							=       []  # this list will be returns.

						# actual sieve of erathostenes
						for i in range(len(__UpperCAmelCase  )  ):
												for j in range(i + 1      ,   len(__UpperCAmelCase  )  ):
																		if (begin_list[i] != 0) and (begin_list[j] % begin_list[i] == 0):
																								__magic_name__:	Optional[Any]							=       0

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

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

						return ans



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

						__magic_name__:	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(__UpperCAmelCase  ):
																		ans.append(__UpperCAmelCase  )

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

						return ans



def   a    (     __UpperCAmelCase      :    Optional[Any]  )    ->							Tuple:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ) and number >= 0, "'number' must been an int and >= 0"

						__magic_name__:	Optional[int]							=       []  # this list will be returns of the function.

						# potential prime number factors.

						__magic_name__:	int							=       2

						__magic_name__:	List[str]							=       number

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

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

						else:
												ans.append(__UpperCAmelCase  )

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

						return ans



def   a    (     __UpperCAmelCase      :    Optional[int]  )    ->							Union[str, Any]:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						__magic_name__:	Optional[Any]							=       0

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

						__magic_name__:	Any							=       max(__UpperCAmelCase  )

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

						return ans



def   a    (     __UpperCAmelCase      :    List[Any]  )    ->							int:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ) and (
						    number >= 0
						), "'number' bust been an int and >= 0"

						__magic_name__:	Dict							=       0

						# prime factorization of 'number'
						__magic_name__:	str							=       prime_factorization(__UpperCAmelCase  )

						__magic_name__:	int							=       min(__UpperCAmelCase  )

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

						return ans



def   a    (     __UpperCAmelCase      :    Dict  )    ->							int:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ), "'number' must been an int"
						assert isinstance(number % 2 == 0      ,   __UpperCAmelCase  ), "compare bust been from type bool"

						return number % 2 == 0



def   a    (     __UpperCAmelCase      :    int  )    ->							Any:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ), "'number' must been an int"
						assert isinstance(number % 2 != 0      ,   __UpperCAmelCase  ), "compare bust been from type bool"

						return number % 2 != 0



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

						__magic_name__:	Any							=       []  # this list will returned

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

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

						# exit variable. for break up the loops
						__magic_name__:	Any							=       True

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

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

																		j += 1

												i += 1

						# precondition
						assert (
						    isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    and (len(__UpperCAmelCase  ) == 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    (     __UpperCAmelCase      :    Optional[int]      ,   __UpperCAmelCase      :    Dict  )    ->							List[str]:
						assert (
						    isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    and isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    and (numbera >= 0)
						    and (numbera >= 0)
						), "'number1' and 'number2' must been positive integer."

						__magic_name__:	Union[str, Any]							=       0

						while numbera != 0:
												__magic_name__:	Dict							=       numbera % numbera
												__magic_name__:	Optional[Any]							=       numbera
												__magic_name__:	str							=       rest

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

						return numbera



def   a    (     __UpperCAmelCase      :    Optional[Any]      ,   __UpperCAmelCase      :    List[Any]  )    ->							Dict:
						assert (
						    isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    and isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    and (numbera >= 1)
						    and (numbera >= 1)
						), "'number1' and 'number2' must been positive integer."

						__magic_name__:	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'
												__magic_name__:	Union[str, Any]							=       prime_factorization(__UpperCAmelCase  )
												__magic_name__:	List[str]							=       prime_factorization(__UpperCAmelCase  )

						elif numbera == 1 or numbera == 1:
												__magic_name__:	Optional[Any]							=       []
												__magic_name__:	Tuple							=       []
												__magic_name__:	List[Any]							=       max(__UpperCAmelCase      ,   __UpperCAmelCase  )

						__magic_name__:	Any							=       0
						__magic_name__:	int							=       0

						__magic_name__:	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:
																								__magic_name__:	Dict							=       prime_fac_a.count(__UpperCAmelCase  )
																								__magic_name__:	Tuple							=       prime_fac_a.count(__UpperCAmelCase  )

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

																		else:
																								__magic_name__:	str							=       prime_fac_a.count(__UpperCAmelCase  )

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

																		done.append(__UpperCAmelCase  )

    # iterates through primeFac2
						for n in prime_fac_a:
												if n not in done:
																		__magic_name__:	Optional[Any]							=       prime_fac_a.count(__UpperCAmelCase  )

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

																		done.append(__UpperCAmelCase  )

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

						return ans



def   a    (     __UpperCAmelCase      :    str  )    ->							List[Any]:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ) and (n >= 0), "'number' must been a positive int"

						__magic_name__:	int							=       0
						__magic_name__:	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(__UpperCAmelCase  ):
																		ans += 1

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

						return ans



def   a    (     __UpperCAmelCase      :    Dict      ,   __UpperCAmelCase      :    str  )    ->							Union[str, Any]:
						assert (
						    is_prime(__UpperCAmelCase  ) and is_prime(__UpperCAmelCase  ) and (p_number_a < p_number_a)
						), "The arguments must been prime numbers and 'pNumber1' < 'pNumber2'"

						__magic_name__:	int							=       p_number_a + 1  # jump to the next number

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

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

						while number < p_number_a:
												ans.append(__UpperCAmelCase  )

												number += 1

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

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

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



def   a    (     __UpperCAmelCase      :    Union[str, Any]  )    ->							str:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ) and (n >= 1), "'n' must been int and >= 1"

						__magic_name__:	Optional[int]							=       []  # will be returned.

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

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

						return ans



def   a    (     __UpperCAmelCase      :    Optional[int]  )    ->							List[Any]:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ) and (
						    number > 1
						), "'number' must been an int and >= 1"

						__magic_name__:	Optional[int]							=       get_divisors(__UpperCAmelCase  )

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

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



def   a    (     __UpperCAmelCase      :    Optional[Any]      ,   __UpperCAmelCase      :    Union[str, Any]  )    ->							Any:
						assert (
						    isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    and isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    and (denominator != 0)
						), "The arguments must been from type int and 'denominator' != 0"

						# build the greatest common divisor of numerator and denominator.
						__magic_name__:	Optional[int]							=       gcd(abs(__UpperCAmelCase  )      ,   abs(__UpperCAmelCase  )  )

						# precondition
						assert (
						    isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  )
						    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    (     __UpperCAmelCase      :    Optional[Any]  )    ->							List[Any]:
						assert isinstance(__UpperCAmelCase      ,   __UpperCAmelCase  ) and (n >= 0), "'n' must been a int and >= 0"

						__magic_name__:	Optional[Any]							=       1  # this will be return.

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

						return ans



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

						__magic_name__:	Union[str, Any]							=       0
						__magic_name__:	Optional[Any]							=       1
						__magic_name__:	Tuple							=       1  # this will be return

						for _ in range(n - 1  ):
												__magic_name__:	List[str]							=       ans
												ans += fiba
												__magic_name__:	List[str]							=       tmp

						return ans











"""simple docstring"""





from typing import List

from .keymap import KEYMAP, get_character



def   a    (     __UpperCAmelCase      :    str  )    ->							Optional[int]:

						def decorator(__UpperCAmelCase      :    List[str]  ):
												__magic_name__:	Union[str, Any]							=       getattr(__UpperCAmelCase      ,   """handle_key"""      ,   []  )
												handle += [key]
												setattr(__UpperCAmelCase      ,   """handle_key"""      ,   __UpperCAmelCase  )
												return func

						return decorator



def   a    (     *__UpperCAmelCase      :    List[str]  )    ->							Union[str, Any]:

						def decorator(__UpperCAmelCase      :    Tuple  ):
												__magic_name__:	Union[str, Any]							=       getattr(__UpperCAmelCase      ,   """handle_key"""      ,   []  )
												handle += keys
												setattr(__UpperCAmelCase      ,   """handle_key"""      ,   __UpperCAmelCase  )
												return func

						return decorator



class       __A		(   SCREAMING_SNAKE_CASE_			):

					def __new__(			cls  :      Optional[Any]     ,		__snake_case  :      List[str]     ,		__snake_case  :      Optional[Any]     ,		__snake_case  :      Any					)			->							List[Any]:
											__magic_name__:	str							=       super().__new__(cls     ,		__snake_case     ,		__snake_case     ,		__snake_case					)
											if not hasattr(__snake_case     ,		"""key_handler"""					):
																	setattr(__snake_case     ,		"""key_handler"""     ,		{}					)
											setattr(__snake_case     ,		"""handle_input"""     ,		KeyHandler.handle_input					)

											for value in attrs.values():
																	__magic_name__:	List[str]							=       getattr(__snake_case     ,		"""handle_key"""     ,		[]					)
																	for key in handled_keys:
																							__magic_name__:	Any							=       value
											return new_cls






					@staticmethod
					def   lowerCamelCase__   (			cls  :      List[Any]					)			->							Optional[Any]:
											__magic_name__:	List[str]							=       get_character()
											if char != KEYMAP["undefined"]:
																	__magic_name__:	Union[str, Any]							=       ord(__snake_case					)
											__magic_name__:	Dict							=       cls.key_handler.get(__snake_case					)
											if handler:
																	__magic_name__:	List[str]							=       char
																	return handler(cls					)
											else:
																	return None



def   a    (     cls      :    List[Any]  )    ->							int:
						return KeyHandler(cls.__name__      ,   cls.__bases__      ,   cls.__dict__.copy()  )



