$(10$pt$)$ A beam of parallel light rays from a laser is incident on a solid transparent sphere of index of refraction $n.$

$($a$)$ If a point image is produced at the back of the sphere, what is the index of refraction of the sphere?

$($b$)$ What index of refraction, if any, will produce a point image at the center of the sphere? Assume that the size of the beam is much smaller than the sphere, and therefore the incident angle is small $(1).$