Trigonometry · Superchallenge problem

Two rays leave a point $B$ at an angle of $30^\circ$. Point $C_0$ lies on the first ray with $BC_0 = 1$. From $C_0$ drop a perpendicular to the second ray, meeting it at $D_1$; from $D_1$ drop a perpendicular back to the first ray, meeting it at $C_1$; from $C_1$ drop a perpendicular to the second ray, and so on forever. Find the total length of the zigzag $C_0D_1 + D_1C_1 + C_1D_2 + \cdots$.