HMMT 2014 G5

Let $\mathcal{C}$ be a circle in the $x y$ plane with radius 1 and center $(0,0,0)$, and let $P$ be a point in space with coordinates $(3,4,8)$. Find the largest possible radius of a sphere that is contained entirely in the slanted cone with base $\mathcal{C}$ and vertex $P$.