Advanced Number Theory · Challenge problem

A sequence is defined by $a_0 = 3$ and $a_{n+1} = a_n^2 - 2$ for $n \ge 0$, so it begins $3, 7, 47, 2207, \ldots$ Find the remainder when the product $a_0 a_1 a_2 \cdots a_9$ is divided by $1000$.