
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 the sequence $\{b_n\}$ is a geometric sequence with a common ratio greater than zero, and $a_1=b_1=2$, $a_3=b_3=8$.
$(1)$ Find the general formula for sequences $\{a_n\}$ and $\{b_n\}$;
$(2)$ Let $c_n=a_{b_n}$, 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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