
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 $T$ of $\{1, 2, 3, \ldots, 20\}$ have the following characteristics?
$(1)$ No two consecutive integers belong to $T$.
$(2)$ If $T$ contains $m$ elements, then $T$ contains no number less than $m+2$.

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