
from math import sqrt






def    lowerCamelCase__				(				__lowerCAmelCase  :  int				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	True

			# 0 and 1 are none primes.
			if number <= 1:
						lowerCAmelCase_						=	False

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

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

			return status






def    lowerCamelCase__				(				__lowerCAmelCase  :  Tuple				):




			"""simple docstring"""


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

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

			lowerCAmelCase_						=	[]  # this list will be returns.

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

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

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  List[str]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	[]

			# 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(snake_case__				):
									ans.append(snake_case__				)

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  Union[str, Any]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	[]  # this list will be returns of the function.

			# potential prime number factors.

			lowerCAmelCase_						=	2

			lowerCAmelCase_						=	number

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

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

			else:
						ans.append(snake_case__				)

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  Optional[Any]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	0

			# prime factorization of 'number'
			lowerCAmelCase_						=	prime_factorization(snake_case__				)

			lowerCAmelCase_						=	max(snake_case__				)

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  Union[str, Any]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	0

			# prime factorization of 'number'
			lowerCAmelCase_						=	prime_factorization(snake_case__				)

			lowerCAmelCase_						=	min(snake_case__				)

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  List[str]				):




			"""simple docstring"""


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

			return number % 2 == 0






def    lowerCamelCase__				(				__lowerCAmelCase  :  str				):




			"""simple docstring"""


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

			return number % 2 != 0






def    lowerCamelCase__				(				__lowerCAmelCase  :  Optional[int]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	[]  # this list will returned

			# creates a list of prime numbers between 2 up to 'number'
			lowerCAmelCase_						=	get_prime_numbers(snake_case__				)
			lowerCAmelCase_						=	len(snake_case__				)

			# run variable for while-loops.
			lowerCAmelCase_						=	0
			lowerCAmelCase_						=	None

			# exit variable. for break up the loops
			lowerCAmelCase_						=	True

			while i < len_pn and loop:
						lowerCAmelCase_						=	i + 1

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

									j += 1

						i += 1

			# precondition
			assert (
			    isinstance(snake_case__    ,							snake_case__				)
			    and (len(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    lowerCamelCase__				(				__lowerCAmelCase  :  List[str]    ,							__lowerCAmelCase  :  Optional[Any]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	0

			while numbera != 0:
						lowerCAmelCase_						=	numbera % numbera
						lowerCAmelCase_						=	numbera
						lowerCAmelCase_						=	rest

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

			return numbera






def    lowerCamelCase__				(				__lowerCAmelCase  :  List[Any]    ,							__lowerCAmelCase  :  int				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	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_						=	prime_factorization(snake_case__				)
						lowerCAmelCase_						=	prime_factorization(snake_case__				)

			elif numbera == 1 or numbera == 1:
						lowerCAmelCase_						=	[]
						lowerCAmelCase_						=	[]
						lowerCAmelCase_						=	max(snake_case__    ,							snake_case__				)

			lowerCAmelCase_						=	0
			lowerCAmelCase_						=	0

			lowerCAmelCase_						=	[]  # 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_						=	prime_fac_a.count(snake_case__				)
												lowerCAmelCase_						=	prime_fac_a.count(snake_case__				)

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

									else:
												lowerCAmelCase_						=	prime_fac_a.count(snake_case__				)

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

									done.append(snake_case__				)

    # iterates through primeFac2
			for n in prime_fac_a:
						if n not in done:
									lowerCAmelCase_						=	prime_fac_a.count(snake_case__				)

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

									done.append(snake_case__				)

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  str				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	0
			lowerCAmelCase_						=	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(snake_case__				):
									ans += 1

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  Any    ,							__lowerCAmelCase  :  List[Any]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	p_number_a + 1  # jump to the next number

			lowerCAmelCase_						=	[]  # this list will be returns.

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

			while number < p_number_a:
						ans.append(snake_case__				)

						number += 1

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

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

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






def    lowerCamelCase__				(				__lowerCAmelCase  :  int				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	[]  # will be returned.

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

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  Optional[Any]				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	get_divisors(snake_case__				)

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

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






def    lowerCamelCase__				(				__lowerCAmelCase  :  Tuple    ,							__lowerCAmelCase  :  List[str]				):




			"""simple docstring"""


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

			# build the greatest common divisor of numerator and denominator.
			lowerCAmelCase_						=	gcd(abs(snake_case__				)    ,							abs(snake_case__				)				)

			# precondition
			assert (
			    isinstance(snake_case__    ,							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    lowerCamelCase__				(				__lowerCAmelCase  :  Any				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	1  # this will be return.

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

			return ans






def    lowerCamelCase__				(				__lowerCAmelCase  :  str				):




			"""simple docstring"""


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

			lowerCAmelCase_						=	0
			lowerCAmelCase_						=	1
			lowerCAmelCase_						=	1  # this will be return

			for _ in range(n - 1				):
						lowerCAmelCase_						=	ans
						ans += fiba
						lowerCAmelCase_						=	tmp

			return ans








from __future__ import annotations

__UpperCAmelCase          =       {
    '''A''': ['''B''', '''C''', '''E'''],
    '''B''': ['''A''', '''D''', '''E'''],
    '''C''': ['''A''', '''F''', '''G'''],
    '''D''': ['''B'''],
    '''E''': ['''A''', '''B''', '''D'''],
    '''F''': ['''C'''],
    '''G''': ['''C'''],
}



class 				lowerCAmelCase_		:
       def __init__(			self,			SCREAMING_SNAKE_CASE_,			SCREAMING_SNAKE_CASE_       )     ->		None:
          UpperCamelCase    :  Union[str, Any]							=	graph
          # mapping node to its parent in resulting breadth first tree
          UpperCamelCase    :  dict[str, str | None]							=	{}
          UpperCamelCase    :  Union[str, Any]							=	source_vertex
       def 					snake_case_       (			self       )     ->		None:
          UpperCamelCase    :  str							=	{self.source_vertex}
          UpperCamelCase    :  str							=	None
          UpperCamelCase    :  int							=	[self.source_vertex]  # first in first out queue

          while queue:
             UpperCamelCase    :  Dict							=	queue.pop(0       )
             for adjacent_vertex in self.graph[vertex]:
                if adjacent_vertex not in visited:
                   visited.add(SCREAMING_SNAKE_CASE_       )
                   UpperCamelCase    :  Tuple							=	vertex
                   queue.append(SCREAMING_SNAKE_CASE_       )






       def 					snake_case_       (			self,			SCREAMING_SNAKE_CASE_       )     ->		str:
          if target_vertex == self.source_vertex:
             return self.source_vertex

          UpperCamelCase    :  Optional[Any]							=	self.parent.get(SCREAMING_SNAKE_CASE_       )
          if target_vertex_parent is None:
             UpperCamelCase    :  Union[str, Any]							=	(
                 F"""No path from vertex: {self.source_vertex} to vertex: {target_vertex}"""
             )
             raise ValueError(SCREAMING_SNAKE_CASE_       )

          return self.shortest_path(SCREAMING_SNAKE_CASE_       ) + F"""->{target_vertex}"""


if __name__ == "__main__":
    __UpperCAmelCase          =       Graph(graph, '''G''')
    g.breath_first_search()
    print(g.shortest_path('''D'''))
    print(g.shortest_path('''G'''))
    print(g.shortest_path('''Foo'''))



