
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]


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Given that $\{a_n\}$ is an arithmetic sequence and $\{b_n\}$ is a geometric sequence, where $(a_1=b_1=1)$, $(a_2+b_3=a_4)$, and $(a_3+b_4=a_7)$.

(I) Find the general term formulas for the sequences $\{a_n\}$ and $\{b_n\}$;

(II) Let $\{c_n=\frac{1}{n}(a_1+a_2+\cdots+a_n)(b_1+b_2+\cdots+b_n)\}$, 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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