ISL 2021 A8

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a))=f\left(a b^{2}+b c^{2}+c a^{2}\right)-f\left(a^{2} b+b^{2} c+c^{2} a\right)$$ for all real numbers $a, b, c$.