Circular Slice II

A random angle $\theta_1 \sim \text{Unif}(0,2\pi)$ is selected. Then, the arc of the unit circle that sweeps out $\theta_1$ radians is marked red going counterclockwise starting from $(1,0)$. Two other angles $\theta_2, \alpha \sim \text{Unif}(0,2\pi)$ IID are also selected. Afterwards, an arc of length $\theta_2$ radians starting from the point that is $\alpha$ radians counterclockwise of the origin is swept out and colored blue. When the blue and red regions intersect, they form a purple region. Given that there is at least one purple region, find the probability there is exactly one purple region.