CZ-SK-MO 1955 Problem 2

Let $\mathsf{S}_1,\mathsf{S}_2$ be concentric spheres with radii $a,b$ respectively, where $a<b.$ Denote $ABCDA'B'C'D'$ a square cuboid ( $ABCD,A'B'C'D$ are the squares and $AA'\parallel BB'\parallel CC'\parallel DD'$ ) such that $A,B,C,D\in\mathsf{S}_2$ and the plane $A'B'C'D'$ is tangent to $\mathsf{S}_1.$ Finally assume that $$\frac{AB}{AA'}=\frac ab.$$ Compute the lengths $AB,AA'.$ How many of such cuboids exist (up to a congruence)?