
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 that the sequence $\{a_n\}$ is an arithmetic sequence with a positive common difference, and the sequence $\{b_n\}$ is a geometric sequence. Moreover, it is known that $a_1 = 1$, $b_1 = 2$, $b_2 - a_2 = 1$, and $a_3 + b_3 = 13$.
(I) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
(II) Let $c_n = a_n \cdot b_n$. Find the sum of the first $n$ terms $S_n$ for the sequence $\{c_n\}$.

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