A uniform solid disk of mass \(M\) and radius \(R\) (rotational inertia \(I = \dfrac{1}{2}MR^2\)) rotates horizontally with angular speed \(\omega_0\) about a fixed frictionless vertical axle through its center. A child of mass \(m = \dfrac{1}{2}M\), modeled as a point mass, runs tangentially along the perimeter of the disk in the direction of rotation with linear speed \(v_0 = 2\omega_0 R\) and steps onto the rim of the disk, sticking to it. Which of the following correctly identifies the final angular speed \(\omega_f\) of the disk-child system and the ratio of the system's final kinetic energy to its initial kinetic energy, \(\dfrac{K_f}{K_i}\)?