HMMT 2025 G4

A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.

Semicircle $S_{1}$ is inscribed in a semicircle $S_{2}$ , which is inscribed in another semicircle $S_{3}$ . The radii of $S_{1}$ and $S_{3}$ are 1 and 10, respectively, and the diameters of $S_{1}$ and $S_{3}$ are parallel. The endpoints of the diameter of $S_{3}$ are $A$ and $B$ , and $S_{2}$ 's arc is tangent to $AB$ at $C$ . Compute $AC \cdot CB$ .