HMMT 2022 T7

Find, with proof, all functions $f: \mathbb{R} \backslash\{0\} \rightarrow \mathbb{R}$ such that

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

for all real $x, y, z$ such that $x y z=1$.