
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 the sequence $\{a_n\}$ with the general term formula $a_n = 3^{n-1}$, in the arithmetic sequence $\{b_n\}$, $b_n > 0$ ($n \in \mathbb{N}^*$), and $b_1 + b_2 + b_3 = 15$, also $a_1 + b_1$, $a_2 + b_2$, $a_3 + b_3$ form a geometric sequence.

(1) Find the general term formula of the sequence $\{b_n\}$;  
(2) Find the sum of the first $n$ terms $T_n$ of the sequence $\{a_n \cdot b_n\}$.

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