A single thin converging lens has focal length \(f\). An object is placed at a distance \(d_o = 3f\) to the left of the lens, forming a real image on the right side of the lens. A second thin converging lens with focal length \(f_0\) is then placed in direct, seamless contact with the first lens. With the object kept at the same position, the distance of the new real image from the combined lens system is half the original image distance. What is the focal length \(f_0\) of the second lens in terms of \(f\)?