
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 sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, for any $n \in \mathbb{N}^*$, the point $(n, S_n)$ lies on the graph of the function $f(x) = 2x^2 - x$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $b_n = \frac{2n^2 - n}{p}$, and the sequence $\{b_n\}$ is an arithmetic sequence, find the value of the non-zero constant $p$.  
(3) Let $c_n = \frac{2n^2 - n}{10}$, $T_n$ is the sum of the first $n$ terms of the sequence $\{c_n\}$, find the smallest positive integer $m$ such that $T_n < m$ holds for all $n \in \mathbb{N}^*$.

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