
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 the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, and the point $\left( n,\frac{S_n}{n} \right)$ (for $n\in \mathbb{N}^*$) all lie on the graph of the function $y=3x-2$.

$(1)$ Find the general formula for the sequence $\{a_n\}$.

$(2)$ Let $b_n=\frac{3}{a_n\cdot a_{n+1}}$, 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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