Trigonometry · Superchallenge problem

Start with a segment $OA_0$ of length $1$. Build right triangle $OA_0A_1$ with the right angle at $A_0$ and $\angle A_0OA_1 = 30^\circ$. Then build right triangle $OA_1A_2$ on the hypotenuse $OA_1$, with the right angle at $A_1$ and $\angle A_1OA_2 = 30^\circ$, turning in the same direction, and keep going. The triangles do not overlap until the segment $OA_n$ first lands back on the ray $OA_0$. Find the length $OA_n$ at that moment.