
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]


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Given the quadratic function $f(x)=3x^{2}-2x$, if the sum of the first $n$ terms of the sequence $\{a_n\}$ is $s_n$, and the point $(n,s_n) (n\in\mathbb{N}_{+})$ all lie on the graph of the function $y=f(x)$.

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

$(2)$ Let $b_n= \dfrac{3}{a_na_{n+1}}$, and $T_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the minimum positive integer value of $m$ such that $T_n < \dfrac{m}{20}$ holds for all $n$ $(n\in\mathbb{N}_{+})$.

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