BALKAN-SL 2021 N1

Let $n \geqslant 3$ be an integer and let

$$M=\left\{\frac{a_{1}+a_{2}+\cdots+a_{k}}{k}: 1 \leqslant k \leqslant n \text { and } 1 \leqslant a_{1}<\cdots<a_{k} \leqslant n\right\}$$

be the set of the arithmetic means of the elements of all non-empty subsets of $\{1,2, \ldots, n\}$. Find $\min \{|a-b|: a, b \in M$ with $a \neq b\}$.