In the accompanying figure, semicircles with centers$A$ and $B$ have radii $4$ and $2$, respectively. Furthermore, they are internally tangent to the circle of diameter $PQ$. Also the semicircles with centers $A$ and $B$ are externally tangent to each other. The circle with center $C$ is internally tangent to the semicircle with diameter $PQ$ and externally tangent to the others two semicircles. Determine the value of the radius of the circle with center $C$.