
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
Consider a sequence $\{a\_n\}$ where all terms are positive, $a\_1=1$, and $S\_n$ is the sum of the first $n$ terms of the sequence. For any $n \in \mathbb{N}^*, 6S\_n=a\_n^2+3a\_n+2$.

1. Find the general term formula for the sequence $\{a\_n\}$.
2. Let $b\_n=\frac{2S\_n}{3n-1}\cdot 2^n$, find the sum of the first $n$ terms of the sequence $\{b\_n\}$, denoted as $T\_n$.

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