CGMO 2022 Problem 6

Find all integers $n$ satisfying the following property. There exist nonempty finite integer sets $A$ and $B$ such that for any integer $m$ , exactly one of these three statements below is true: (a) There is $a \in A$ such that $m \equiv a \pmod n$ , (b) There is $b \in B$ such that $m \equiv b \pmod n$ , and (c) There are $a \in A$ and $b \in B$ such that $m \equiv a + b \pmod n$ .