
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
It is known that all terms of the sequence $\{a_n\}$ are positive numbers, and for any $n\in N^*$, the sum of its first $n$ terms $S_n$ satisfies $S_n= \frac{1}{6}(a_n+1)(a_n+2)$, and $a_2$, $a_4$, $a_9$ form a geometric sequence. The general formula for the sequence $\{a_n\}$ is.

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