Find the dimensions of the rectangular box with maximum volume in the first octant with one vertex at the origin and the opposite vertex on the ellipsoid 4x2 + 4y? + z? = 4. The length of the box in the x-direction is. (Type an exact answer, using radicals as needed.) The width of the box in the y-direction is (Type an exact answer, using radicals as needed.) The height of the box in the z-direction is (Type an exact answer, using radicals as needed.)