Consider a right circular cone of base radius $4$ cm and height $10$ cm. A cylinder is to be placed inside the cone with one of the flat surfaces resting on the base of the cone. Find the largest possible total surface area (in sq cm) of the cylinder.

$\frac{100 \pi}{3}$

$\frac{80 \pi}{3}$

$\frac{120 \pi}{7}$

$\frac{130 \pi}{9}$

$\frac{110 \pi}{7}$