






"""simple docstring"""





A__					: Union[str, Any]=	tuple[float, float, float]
A__					: Dict=	tuple[float, float, float]







def   lowerCAmelCase_( SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE					)					->	Vectorad:
    """simple docstring"""

    UpperCamelCase__         =    end_pointa[0] - end_pointa[0]
    UpperCamelCase__         =    end_pointa[1] - end_pointa[1]
    UpperCamelCase__         =    end_pointa[2] - end_pointa[2]
    return (x, y, z)







def   lowerCAmelCase_( SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE					)					->	Vectorad:
    """simple docstring"""

    UpperCamelCase__         =    ab[1] * ac[2] - ab[2] * ac[1]  # *i
    UpperCamelCase__         =    (ab[0] * ac[2] - ab[2] * ac[0]) * -1  # *j
    UpperCamelCase__         =    ab[0] * ac[1] - ab[1] * ac[0]  # *k
    return (x, y, z)







def   lowerCAmelCase_( SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE					)					->	bool:
    """simple docstring"""

    return tuple(round(SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE					) for x in vector					) == (0, 0, 0)







def   lowerCAmelCase_( SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE = 10					)					->	bool:
    """simple docstring"""

    UpperCamelCase__         =    create_vector(SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE					)
    UpperCamelCase__         =    create_vector(SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE					)
    return is_zero_vector(get_ad_vectors_cross(SCREAMING_SNAKE_CASE ,	SCREAMING_SNAKE_CASE					) ,	SCREAMING_SNAKE_CASE					)












"""simple docstring"""





def   lowerCAmelCase_( SCREAMING_SNAKE_CASE					)					->	bool:
    """simple docstring"""

    UpperCamelCase__         =    (1 + 24 * n) ** 0.5
    return ((1 + root) / 6) % 1 == 0







def   lowerCAmelCase_( SCREAMING_SNAKE_CASE = 50_00					)					->	int:
    """simple docstring"""

    UpperCamelCase__         =    [(i * (3 * i - 1)) // 2 for i in range(1 ,	SCREAMING_SNAKE_CASE					)]
    for i, pentagonal_i in enumerate(SCREAMING_SNAKE_CASE					):
        for j in range(SCREAMING_SNAKE_CASE ,	len(SCREAMING_SNAKE_CASE					)					):
            UpperCamelCase__         =    pentagonal_nums[j]
            UpperCamelCase__         =    pentagonal_i + pentagonal_j
            UpperCamelCase__         =    pentagonal_j - pentagonal_i
            if is_pentagonal(SCREAMING_SNAKE_CASE					) and is_pentagonal(SCREAMING_SNAKE_CASE					):
                return b

    return -1


if __name__ == "__main__":
  print(F"""{solution() = }""")




