HMMT 2021 U20

Let $f(x)=x^{3}-3 x$. Compute the number of positive divisors of

$$\left\lfloor f\left(f\left(f\left(f\left(f\left(f\left(f\left(f\left(\frac{5}{2}\right)\right)\right)\right)\right)\right)\right)\right)\right\rfloor$$

where $f$ is applied 8 times.