
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 $f(x)=a\ln x-ax-3$ where $a\in \mathbb{R}$.  
$(1)$ Find the intervals where the function $f(x)$ is monotonic.  
$(2)$ If the slope of the tangent line to the graph of the function $f(x)$ at the point $(2,f(2))$ is $\frac{1}{2}$, let $g(x)=f(x)-\frac{m}{x}$. If the function $g(x)$ is monotonically increasing on the interval $[1,2]$, find the range of real numbers for $m$.

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