Question 59
Let $ABC$ be a triangle, and let points $P$ and $Q$ lie on $BC$ such that $P$ is closer to $B$ than $Q$ is. Suppose that the radii of the incircles of triangles $ABP$, $APQ$, and $AQC$ are all equal to $1$, and that the radii of the corresponding excircles opposite $A$ are $3$, $6$, and $5$, respectively. If the radius of the incircle of triangle $ABC$ is $\frac{3}{2}$, find the radius of the excircle of triangle $ABC$ opposite $A$.