Školjka

Let $n, m, k$ and $l$ be positive integers with $n \neq 1$ such that $n^k + mn^l + 1$ divides $n^{k+l} - 1$. Prove that \begin{itemize} \item $m = 1$ and $l = 2k$; or \item $l|k$ and $m = \frac{n^{k-l}-1}{n^l-1}$. \end{itemize}