Find all natural numbers $m,n,l$ such that $$m+n = (n,m)^2; \quad n+l = (n,l)^2; \quad l+m = (m,l)^2$$
where $(a,b)$ is the greatest common divisor of $a$ and $b$.
I only managed to find that if $d = (m,n,l)$ then $d=1$ or $d=2$. But I couldn't prove anything else in each of the cases.