
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
Let $\{a_n\}$ be a sequence where each term is positive, and let $S_n$ be the sum of the first $n$ terms of $\{a_n\}$. Given that $S_n$ satisfies $$2S_n^2 - (3n^2 + 3n - 2)S_n - 3(n^2+n) = 0$$ for $n \in \mathbb{N}^*$.  
(i) Find the value of $a_1$.  
(ii) Determine the general formula for the sequence $\{a_n\}$.  
(iii) Let $b_n = \frac{a_n}{3^{n+1}}$. Calculate the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted by $T_n$.

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