I want to construct a conformal mapping of the region $\{z \in \mathbb{C}: |z - 1/2| < 1 \text{ and } |z + 1/2| < 1 \}$ to the unit disk. I think I would use the Mobius transformation $-\frac{z - \sqrt{3}i/2}{z + \sqrt{3}i/2}$ to map the intersection of the two circles to two lines. Then, somehow I would want to map this to the upper half of the plane, and then use the Cayley transformation to get the unit disk. I'm not sure how to modify this approach and how to get to the upper half of the plane.
Any help is appreciated.