HMIC 2016 Problem 3

Denote by $\mathbb{N}$ the positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w, x, y, z \in \mathbb{N}$,

$$f(f(f(z))) f(w x f(y f(z)))=z^{2} f(x f(y)) f(w)$$

Show that $f(n!) \geq n$ ! for every positive integer $n$.