
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
Given a geometric sequence $\{a_n\}$ with the first term $a_1=\frac{1}{3}$, and the sum of the first n terms denoted as $S_n$, if $S_1$, $2S_2$, and $3S_3$ form an arithmetic sequence, find the general term of $\{a_n\}$.

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