
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 an arithmetic sequence $\{a_n\}$ satisfies $a_{n+1} > a_n$, $a_1 = 1$, and by adding 1, 1, 3, respectively, to the first three terms, they become the first three terms of a geometric sequence $\{b_n\}$.

(1) Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$.
(2) Let $c_n = a_n \cdot b_n$, find the sum of the first $n$ terms $S_n$ of the sequence $\{c_n\}$.

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