
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 first term of the geometric sequence $\{a_n\}$ is 2, the sum of the first $n$ terms of the arithmetic sequence $\{b_n\}$ is $S_n$, and $a_1+a_2=6$, $2b_1+a_3=b_4$, $S_3=3a_2$.
(I) Find the general term formulas for $\{a_n\}$ and $\{b_n\}$;
(II) Let $c_n=b_{a_n}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$.

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