Junior Mathematical Olympiad 2024 Section B Q16

Let $A$ be the set of $2n$ positive integers $1, 2, 3, \ldots, 2n$, where $n \geq 1$. For which values of $n$ can this be split into $n$ pairs of integers in such a way that every pair has a sum which is a multiple of $3$? As always in Olympiad problems such as this, you also need to explain why no other values of $n$ are possible.