
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 the quadratic function $f(x)=3x^{2}-2x$, the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}$, and the point $(n,S_{n})$ (where $n \in \mathbb{N}^{*}$) is on the graph of the function $y=f(x)$.

$(1)$ Find the general formula for the sequence $\{a_{n}\}$;

$(2)$ Let $b_{n}=\frac{3}{a_{n}a_{n+1}}$, and $T_{n}$ be the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Find the smallest positive integer $m$ such that $T_{n} < \frac{m}{20}$ holds for all $n \in \mathbb{N}^{*}$.

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