
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
How many non-empty subsets \(S\) of \(\{1,2,3,\ldots,10\}\) have the following two properties?
1. No two consecutive integers belong to \(S\).
2. If \(S\) contains \(k\) elements, then \(S\) contains no number less than \(k\).

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