Kinematics Challenging Problem

For "Fast & Furious Race", there is a track as shown below. The track is made up of uniform path segments $AB, BC, CD,\ldots$ each with the same distance $900\text{ m}.$ The radius of the path is $R$ and $h \text{ of } AB= \frac12 h\text{ of } BC=\frac14 h\text{ of } CD=\frac18 h\text{ of } DE=\cdots,$ where $h$ denotes the height and points $A, B, C, D, E$ are all on a vertical line.

Alto K-10 and Ferrari both start from point $A$ simultaneously. Alto K-10 moves with constant speed $\sqrt{10}\text{ m/s}$ and Ferrari starts from rest with uniformly increasing speed. Both cars meet after traveling $3970\text{ m}.$ When both car have the same speed, the magnitude of the relative horizontal velocity is $\alpha\text{ m/s}$ and the ratio of their vertical components of velocities (Alto K-10 to Ferrari) is $\beta.$

Find $\alpha \times \beta.$ (Take suitable apporximation.)