A block of mass \(M\) is attached to an ideal horizontal spring of spring constant \(k\) and oscillates on a frictionless horizontal surface with amplitude \(A_0\).

In Scenario 1, at the instant the block reaches its maximum displacement \(x = A_0\), a piece of mass \(\dfrac{M}{2}\) is gently lifted vertically away with no horizontal force, and the remaining mass oscillates with amplitude \(A_1\).

In Scenario 2, an identical setup begins with the same initial amplitude \(A_0\), but the piece of mass \(\dfrac{M}{2}\) is gently lifted vertically away as the block passes through the equilibrium position \(x = 0\), and the remaining mass oscillates with amplitude \(A_2\).

What is the ratio \(\dfrac{A_1}{A_2}\) of the resulting amplitudes?