
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 $\left\{a_{n}\right\}$ with the sum of its first $n$ terms $S_{n}=-n^{2}+9n+2 \, (n \in \mathbf{N}^{*})$.
1. Find the general term formula for the sequence $\left\{a_{n}\right\}$.
2. Let $b_{n}=\frac{1}{n(12-a_{n})} \, (n \in \mathbf{N}^{*})$ and $T_{n}=b_{1}+b_{2}+\cdots+b_{n}$. Does there exist a smallest positive integer $m$ such that the inequality $T_{n}<\frac{m}{32}$ holds for all positive integers $n$? If so, determine the value of $m$. If not, explain why.

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