When a bright light source is placed $28\mathrm{.}2$ cm in front of a lens, there is an upright image $8\mathrm{.}10$ cm from the lens. There is also a faint inverted image $6\mathrm{.}40$ cm from the lens on the incident$-$light side due to reflection from the front surface of the lens. When the lens is turned around, this weaker, inverted image is $10\mathrm{.}0$ cm in front of the lens.

Determine the index of refraction of the lens, $\backslash ($ n $\backslash ).$