
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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Let $\{a_n\}$ be an arithmetic sequence, $\{b_n\}$ be a geometric sequence with a positive common ratio. Given that $a_1 = b_1 = 3$, $b_2 = a_3$, and $b_3 = 4a_2 + 3$.
(I) Find the general term formulas for $\{a_n\}$ and $\{b_n\}$;
(II) Let $\{c_n\}$ be a sequence such that $c_n = \begin{cases} 1, & \text{if n is odd,} \\ b_{\frac{n}{2}}, & \text{if n is even.}\end{cases}$ Find the value of $a_1c_1 + a_2c_2 + \dots + a_{2n}c_{2n}$ for all positive integers $n$.

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