def METHOD_NAME(pole):
    origin = get_random_orthogonal(pole, rng)
    GreatCircleICRSFrame(pole=pole, origin=origin)
    GreatCircleICRSFrame(pole=pole, origin=origin, priority="pole")
    with pytest.raises(ValueError):
        GreatCircleICRSFrame(pole=pole, ra0=origin.ra)
    # Slightly adjust the origin so it is not orthogonal:
    new_origin = origin.spherical_offsets_by(
        1.23 * u.deg, -2.42 * u.deg
    )  # random values
    with pytest.warns():
        f1 = GreatCircleICRSFrame(pole=pole, origin=new_origin)
    # default priority="origin"
    assert f1.origin.ra == new_origin.ra
    assert f1.origin.dec == new_origin.dec
    assert np.isclose(f1.origin.cartesian.xyz @ f1.pole.cartesian.xyz, 0.0)
    with pytest.warns():
        f2 = GreatCircleICRSFrame(pole=pole, origin=new_origin, priority="pole")
    assert f2.pole.ra == pole.ra
    assert f2.pole.dec == pole.dec
    assert np.isclose(f2.origin.cartesian.xyz @ f2.pole.cartesian.xyz, 0.0)