
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 \( a_1, a_2, a_3, \dots \) be a sequence defined by \( a_1 = 1, a_2 = 1, a_3 = 2 \) and \( a_{n+3} = a_{n+2} + a_{n+1} + a_n \) for \( n \geq 1 \). Find
\[
\sum_{n=1}^\infty \frac{a_n}{3^{n+1}}.
\]

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