Polar Coordinates and Complex Numbers · Challenge problem

The point with polar coordinates $(3, 10^\circ)$ is reflected across the line $\theta = 20^\circ$ (through the pole), the image is reflected across the line $\theta = 65^\circ$, that image across $\theta = 20^\circ$ again, and so on, alternating, for $27$ reflections in all (the first and the last are across $\theta = 20^\circ$). The final point is $(3, \alpha)$ with $0^\circ \le \alpha < 360^\circ$. Find $\alpha$.