HMMT November 2008 U9

Let $a_{0}=\frac{6}{7}$, and

$$a_{n+1}= \begin{cases}2 a_{n} & \text { if } a_{n}<\frac{1}{2} \\ 2 a_{n}-1 & \text { if } a_{n} \geq \frac{1}{2}\end{cases}$$

Find $a_{2008}$.