
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 sequence $\{a\_n\}$ that satisfies $(a_{n+1} - 1)(a_n - 1) = 3(a_n - a_{n+1})$, with $a_1 = 2$, and let $b_n = \frac{1}{a_n - 1}$.

1. Find the general term formula for the sequence $\{b_n\}$.
2. Find the sum of the first $n$ terms, denoted as $S_n$, for the sequence $\{b_n \cdot 3^n\}$.

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