
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
In the sequence $\{a_n\}$ and the geometric sequence $\{b_n\}$, $a_1 = 0$, $a_3 = 2$, and $b_n = 2a_n + 1$ ($n \in \mathbb{N}^*$).

(1) Find the general formula for the sequences $\{b_n\}$ and $\{a_n\}$;  
(2) If $c_n = a_n \cdot b_n$, 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}
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