HMMT 2011 U28

Let $A B C$ be a triangle, and let points $P$ and $Q$ lie on $B C$ such that $P$ is closer to $B$ than $Q$ is. Suppose that the radii of the incircles of triangles $A B P, A P Q$, and $A Q C$ 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 $A B C$ is $\frac{3}{2}$, find the radius of the excircle of triangle $A B C$ opposite $A$.