
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $x+y \equiv 2z \pmod{99}$ . Determine all possible values of  $m$ .

Present the answer in LaTex format: \boxed{Your answer}
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