
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
Let the function $y=f(x)$ have its first derivative $f'(x)$ and its second derivative $f''(x)$ on the interval $(a, b)$. If $f''(x)<0$ holds for all $x$ in $(a,b)$, then the function $f(x)$ is called a "convex function" on $(a,b)$. Given $f(x) = \frac{1}{12}x^4 - \frac{1}{6}mx^3 - \frac{3}{2}x^2$,  
(1) Find $f'(x)$ and $f''(x)$;  
(2) If $f(x)$ is a "convex function" on the interval $(-1, 3)$, determine the value of the real number $m$;  
(3) If for all real numbers $m$ satisfying $|m|\leq2$, the function $f(x)$ is always a "convex function" on $(a, b)$, find the maximum value of $b-a$.

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