TSTST 2013 Problem 6

Let $\mathbb{N}$ be the set of positive integers. Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ that satisfy the equation $$f^{a b c-a}(a b c)+f^{a b c-b}(a b c)+f^{a b c-c}(a b c)=a+b+c$$ for all $a, b, c \geq 2$. (Here $f^{k}$ means $f$ applied $k$ times.)