
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 that the terms of the geometric sequence $\{a\_n\}$ are all positive, and $6a\_2$, $1$, $4a\_1$ form an arithmetic sequence, while $3a\_6$, $a\_3$, $3a\_2$ form a geometric sequence.

1. Find the general term formula for the sequence $\{a\_n\}$;
2. Given that $b\_n = \log\_3 \frac{1}{a\_n}$, let $c\_n = a\_n \cdot b\_n$, find the sum of the first $n$ terms of the sequence $\{c\_n\}$, denoted as $S\_n$.

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