
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 all terms of the sequence $\{a_n\}$ are positive, and the sum of the first $n$ terms is $S_n$.
(1) If for any $n \in \mathbb{N}^*, S_n = \frac{n^2 + n + 1}{2}$, find $a_n$;
(2) If $a_1 = 1, a_2 = 2, b_n = a_{2n-1} + a_{2n}$, and the sequence $\{b_n\}$ is a geometric sequence with common ratio 3, find $S_{2n}$.

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