
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 $\{a_{n}\}$ that satisfies $a_{n+1} > a_{n}$, $a_{1}=1$, and the first three terms of this sequence become the first three terms of a geometric sequence $\{b_{n}\}$ when added to $1$, $1$, $3$ respectively.

1. Find the general term formulas for the sequences $\{a_{n}\}$ and $\{b_{n}\}$.
2. Let $c_{n} = a_{n} \cdot b_{n}$, find the sum $S_{n}$ of the first $n$ terms of the sequence $\{c_{n}\}$.

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