Circles $A$, $B$ and $C$ are externally tangent to each other, and internally tangent to circle $D$. Circles $A$ and $B$ are congruent. Circle $C$ has radius $\frac{1}{2}$. Circle $D$ has radius $2$. Circle $C$ is tangent to circle $D$ at a point on the x-axis, and its center is on the x-axis. What is the radius of circle $A$?