BALKAN-SL 2008 N2

Let $c$ be a positive integer. The sequence $a_1,a_2,\ldots$ is defined as follows $a_1=c$, $a_{n+1}=a_n^2+a_n+c^3$ for all positive integers $n$. Find all $c$ so that there are integers $k\ge1$ and $m\ge2$ so that $a_k^2+c^3$ is the $m$ t h power of some integer.