
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 $f(x)=\ln x+ax^{2}-3x$.  
$(1)$ If the tangent line to the graph of the function $f(x)$ at the point $(1,f(1))$ is $y=-2$, find the local minimum value of the function $f(x)$.  
$(2)$ If $a=1$, for any $x_{1}$, $x_{2}\in \left[1,2\right]$, when $x_{1} \lt x_{2}$, the inequality $f(x_{1})-f(x_{2}) \gt \frac{{m({{x_2}-{x_1}})}}{{{x_1}{x_2}}}$ always holds, find the range of real number $m$.

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