Middle European Mathematical Olympiad 2023 Problem I-4

Let $n$ and $m$ be positive integers. We call a set $S$ of positive integers $(n, m)$-good if it satisfies the following three conditions:

(i) We have $m \in S$.

(ii) For all $a \in S$, all of the positive divisors of $a$ are elements of $S$ too.

(iii) For all mutually different numbers $a, b \in S$, we have $a^n + b^n \in S$.

Determine all pairs $(n, m)$ such that the set of all positive integers is the only $(n, m)$-good set.