
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 $\mathcal{S}$ be the smallest subset of the set of integers that satisfies the following properties:
1) $0 \in \mathcal{S}$
2) If $x \in \mathcal{S}$, then $3x \in \mathcal{S}$ and $3x+1 \in \mathcal{S}$.

Find the number of non-negative integers in the set $\mathcal{S}$ that do not exceed 2009.

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