
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 of positive numbers $\{a_n\}$ with the sum of the first $n$ terms being $S_n$, $a_1 > 1$, and $6S_n = a_n^2 + 3a_n + 2$ for $n\in\mathbb{N}^*$.
(Ⅰ) Find the general term $a_n$ of the sequence $\{a_n\}$.
(Ⅱ) If $b_n = \frac{a_n - 1}{2^n}$, find the sum of the first $n$ terms $T_n$ of the sequence.

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