IMC 2025 Problem 7

Let $\mathbb{Z}_{>0}$ be the set of positive integers. Find all nonempty subsets $M \subseteq \mathbb{Z}_{>0}$ satisfying both of the following properties: a) if $x \in M$ , then $2 x \in M$ , b) if $x, y \in M$ and $x+y$ is even, then $\frac{x+y}{2} \in M$ .