
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
Determine how many non-empty subsets $S$ of $\{1, 2, 3, \ldots, 10\}$ satisfy the following properties:
1. No three consecutive integers belong to $S$.
2. If $S$ contains $k$ elements, then $S$ contains no number less than $2k$.

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