
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 the derivative $f'(x)$ on the interval $(a,b)$, and $f'(x)$ has the derivative $f''(x)$ on $(a,b)$. If $f''(x) < 0$ holds true for all $x$ in $(a,b)$, then the function $f(x)$ is called a "convex function" on $(a,b)$. It is given that $f(x)= \frac{1}{12}x^4 - \frac{1}{6}mx^3 - \frac{3}{2}x^2$.

(Ⅰ) If $f(x)$ is a "convex function" on the interval $(-1,3)$, determine the value of the real number $m$;  
(Ⅱ) If $f(x)$ is a "convex function" on $(a,b)$ for all real numbers $m$ such that $|m| \leqslant 2$, find the maximum value of $b-a$.

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