BALKAN 2008 Problem 4

Let $c$ be a positive integer. The sequence $a_{1}, a_{2}, \ldots, a_{n}, \ldots$ is defined by $a_{1}=c$, and $a_{n+1}=a_{n}^{2}+a_{n}+c^{3}$, for every positive integer $n$. Find all values of $c$ for which there exist some integers $k \geq 1$ and $m \geq 2$, such that $a_{k}^{2}+c^{3}$ is the $m^{\text {th }}$ power of some positive integer.