A block of mass \(M\) is attached to an ideal horizontal spring of force constant \(k\) and oscillates on a frictionless horizontal surface with an initial amplitude \(A_0\) and total mechanical energy \(E_0\). At the exact instant the block passes through its equilibrium position (\(x = 0\)), a small lump of sticky clay of mass \(m\) is dropped vertically onto the block from a negligible height and immediately sticks to it. Which of the following correctly gives the new amplitude of oscillation \(A_f\) and the ratio of the new total mechanical energy to the original mechanical energy \(\dfrac{E_f}{E_0}\) of the oscillating system?