
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 function $f(x) = ax^2 - 2x + c$, and the solution set of $f(x) > 0$ is $\{x | x \neq \frac{1}{a}\}$.  
(1) Find the minimum value of $f(2)$ and the explicit form of $f(x)$ when $f(2)$ is at its minimum value;  
(2) When $f(2)$ is at its minimum value, it holds that for any $x > 2$, $f(x) + 4 \ge m(x - 2)$ is always true, find the range of the real number $m$.

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