RU-SPB 2020 Problem 92

For the triple $(a,b,c)$ of positive integers we say it is interesting if $c^2+1\mid (a^2+1)(b^2+1)$ but none of the $a^2+1, b^2+1$ are divisible by $c^2+1$ . Let $(a,b,c)$ be an interesting triple, prove that there are positive integers $u,v$ such that $(u,v,c)$ is interesting and $uv<c^3$ .