
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 the function $f(x) = a\ln x - 2ax + 3$ $(a \neq 0)$.  
$(1)$ Find the extreme values of the function $f(x)$.  
$(2)$ If the slope of the tangent line to the graph of the function $y = f(x)$ at $x = 2$ is $\frac{3}{2}$, and $g(x) = \frac{1}{3}x^{3} + x^{2}[f'(x) + m]$ is not a monotonic function on the interval $(1,3)$, and when $x \in (0,1]$, $f(x)$ is not less than $\frac{2}{3}x^{3} - 2m$, find the range of real number $m$.

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