Determine all the functions $f:\mathbb{N} \to \mathbb{N}$ such that $(f(m)+n)(f(n)+m)$ is a perfect square for every $(m,n)$

Question

Determine all the functions $f:\mathbb{N} \to \mathbb{N}$ such that $(f(m)+n)(f(n)+m)$ is a perfect square $\forall m, n \in \mathbb{N}$.

I have been struggling with this problem for a while, any ideas and hints would be appreciated.

It is clear that $(f(m)+n)(f(n)+m)$ is a perfect square when $f(m)+n =f(n)+m$ but that consideration leads to a partial solution to the problem i.e. functions $f(x) = x + c$, where $c \in \mathbb{N}$.

I'm wondering if other types of $f$ could solve this problem. Are they surjective, injective? How can one prove or disprove the existence of such functions?