Let $a_1,$ $a_2,$ $a_3,$ $\dots$ be a sequence of real numbers satisfying \[a_n = a_{n - 1} a_{n + 1}\]for all $n \ge 2.$ If $a_1 = 1 + \sqrt{7}$ and $a_{1776} = 13 + \sqrt{7},$ then determine $a_{2009}.$
a_{2009} is equa to 3 - 2*sqrt(7).
