Expected Steps Between Circle Intersections

Two unit circles are centered at $(0,0)$ and $(1,0)$. They overlap and share exactly two intersection points -- call them $A$ (the upper one) and $B$ (the lower one). A particle starts at $A$. At each step, it picks one of the two circles uniformly at random (prob $\frac{1}{2}$ each), then jumps to a uniformly random point on that circle's boundary. The process stops when the particle lands at $B$. Under the natural discrete interpretation (each step can only land at one of the two intersection points), what is the expected number of steps to reach $B$?