Recurrence relation with variable $x$ in the denominator. A recurrence is defined such that $$f_n(x)=f_1(f_{(n-1)}(x)); x\ge2$$ 
$f_1(x)$ is defined as 
$$f_1(x)= \frac 23 -\frac 3{3x+1}$$
How can I find the values of $x$ for which the following holds true? $$f_{1001}(x)= x-3$$
I was not able to deduce any kind of pattern in the above recurrence. I did try to find a a pattern by finding $f_2$ and $f_3$ but I could not make out any generating function. How should I proceed?

Notice that $$f_2(x) = \frac{2}{3}-\frac{3}{3\cdot\left(\frac{2}{3}-\frac{3}{3x+1}\right)+1} = \frac{2}{3}-\frac{3x+1}{3x-2}$$ 
$$f_3(x) = \frac{2}{3}-\frac{3}{3\cdot\left(\frac{2}{3}-\frac{3x+1}{3x-2}\right)+1} = \frac{2}{3}+\frac{3x-2}{3}$$
$$f_4(x) = \frac{2}{3}-\frac{3}{3\cdot\left(\frac{2}{3}+\frac{3x-2}{3}\right)+1} = \frac{2}{3}-\frac{3}{3x+1}$$
This implies $f_4(x)=f_1(x)$. So we see that this is a cycle of $3$, meaning $$x-3=f_{1001}(x)=f_{998}(x)=\ ...\ =f_2(x)= \frac{2}{3}-\frac{3x+1}{3x-2}\implies x = \frac{5}{3}$$
