
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, and $\{b_n\}$ is a geometric sequence, where $a_1=b_1=1$, $a_2 \neq b_2$, and $b_2$ is the arithmetic mean of $a_1$ and $a_2$, $a_2$ is the arithmetic mean of $b_2$ and $b_3$.

(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.

(2) Let $c_n = \frac{1}{n} \left( a_1+a_2+\cdots+a_n \right) \left( b_1+b_2+\cdots+b_n \right)$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, $S_n$.

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