2012 AIME II Problem 11

Let $f_1(x) = \frac{2}{3}-\frac{3}{3x+1}$, and for $n \ge 2$, define $f_n(x) = f_1(f_{n-1} (x))$. The value of x that satisfies $f_{1001}(x) = x - 3$ can be expressed in the form $\frac{m}{n}$,
where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.