A discrete mathematics problem by Jerry Han Jia Tao

Let \(S\) be a subset of \({0,1,2,\ldots,98} \) with exactly \(m\geq 3\) (distinct) elements, such that for any \(x,y\in S\) there exists \(z\in S\) satisfying \(x+y \equiv 2z \pmod{99}\). Find the sum of all possible values of \(m\).

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