
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 non-empty set with the following properties: ① All elements are positive integers; ② If $x \in S$, then $10-x \in S$.
(1) Provide an example of the set $S$ that meets the conditions and contains one, two, and three elements respectively;
(2) Does there exist a set $S$ with exactly six elements? If it exists, list all such sets $S$; if not, explain why;
(3) Inspired by the solutions to (1) and (2), what general conclusions can we draw about the set $S$? (At least two conclusions are required).

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