
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 two sequences $\{a\_n\}$ and $\{b\_n\}$ with respective sums of the first $n$ terms denoted as $S\_n$ and $T\_n$. It is known that $a\_n > 0$, $(a\_n+1)^2=4(S\_n+1)$, and $b\_nS\_n-1=(n+1)^2$, where $n \in \mathbb{N}^*$.
(1) Find the general term formula for the sequence $\{a\_n\}$;
(2) Find the sum of the first $n$ terms, $T\_n$, for the sequence $\{b\_n\}$.

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