
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 quadratic function $y=f(x)$ whose graph passes through the origin, the solution set of $f(x) < 0$ is $\left( 0,\frac{2}{3} \right)$, and the minimum value of $y=f(x)$ is $-\frac{1}{3}$. The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, and the points $(n,S_n)(n\in \mathbb{N}^*)$ all lie on the graph of the function $y=f(x)$.

(Ⅰ) Find the general formula for the sequence $\{a_n\}$;

(Ⅱ) Let $b_n=\frac{3}{a_na_{n+1}}$, 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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