Points, Lines & Planes · Superchallenge problem

Segment $\overline{AB}$ has length $96$. Let $P_0 = A$ and $P_1 = B$, and for each $n \ge 1$ let $P_{n+1}$ be the midpoint of $\overline{P_{n-1}P_n}$. The points $P_0, P_1, P_2, \ldots$ all lie on $\overline{AB}$ and get closer and closer to a single point $L$. Find $AL$.