MathNet 2013 Question 16

Let $S$ be a subset of $m$ elements of $\{0, 1, 2, \dots, 98\}$, $m \geq 3$, such that for any $x, y \in S$, there exists $z \in S$ with $x + y \equiv 2z \pmod{99}$. Find all possible values of $m$.