A thin uniform \(\operatorname{rod} A B\) has mass \(k M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere with centre \(O\), mass \(k M\) and radius \(2 a\). The end \(B\) of the rod is rigidly attached to the circumference of a uniform ring with centre \(C\), mass \(M\) and radius \(a\). The points \(C, B, A, O\) lie in a straight line. The horizontal axis \(L\) passes through the mid-point of the rod and is perpendicular to the rod and in the plane of the ring (see diagram). The object consisting of the rod, the ring and the hollow sphere can rotate freely about \(L\).
(i) Show that the moment of inertia of the object about \(L\) is \(\frac{3}{2}(8 k+3) M a^{2}\).

The object performs small oscillations about \(L\), with the ring above the sphere as shown in the diagram.
(ii) Find the set of possible values of \(k\) and the period of these oscillations in terms of \(k\).