
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 $\overrightarrow {a}=(x^{2}，-2)$ and $\overrightarrow {b}=(3，x)$, with $f(x)= \overrightarrow {a}\cdot \overrightarrow {b}$. The sequence $\{a_n\}$ has a sum of its first $n$ terms $S_n$, and the point $(n, S_n)$ lies on the graph of $y=f(x)$ for all positive integers $n$.

1. Find the general formula for the sequence $\{a_n\}$.
2. Let $b_n = \frac{3}{a_na_{n+1}}$. Denote $T_n$ as the sum of the first $n$ terms of the sequence $\{b_n\}$. Find the maximum positive integer $m$ such that $T_n > \frac{m}{21}$ for all positive integers $n$.

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