
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 $f(x)=a\ln x-ax-3$ $(a\in \mathbb{R})$.

$(1)$ Find the intervals of monotonicity for the function $f(x)$;

$(2)$ If the tangent line to the graph of $y=f(x)$ at the point $(2,f(2))$ has an angle of inclination of $45^{\circ}$, and for any $t\in[1,2]$, the function ${g}(x)={{x}^{3}}+{{x}^{2}}\cdot [{f}'(x)+\dfrac{m}{2}]$ is not monotonic on the interval $(t,3)$, find the range of values for $m$.

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