
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 a sequence $\{a_n\}$ where each term is positive, and the sum of the first $n$ terms is $S_n$, $a_1 > 1$, and $6S_n = a_n^2 + 3a_n + 2, n \in \mathbb{N}^*$. 

(1) Find the general formula for the sequence $\{a_n\}$.

(2) If $b_n = \frac{a_n - 1}{2^n}$, find the sum of the first $n$ terms of the sequence, $T_n$.

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