
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 the arithmetic sequence $\{a_n\}$ that satisfies $(a_n + 2a_{n+1} = 6n + 1, n \in \mathbb{N}^*)$,

(I) Find the general term formula for the sequence $\{a_n\}$;

(II) If $\{b_n\} = 2a_n\cos^2\left(\frac{n\pi}{2}\right)$, find the sum of the first $2n$ terms of the sequence $\{b_n\}$, denoted as $T_{2n}$.

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