
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 $f(x)=a(x^2+1)+\ln x$.
- $(1)$ Discuss the monotonicity of the function $f(x)$;
- $(2)$ If for any $a\in (-4,-2)$ and $x\in [1,3]$, it always holds that $ma-f(x) > a^2$, find the range of the real number $m$.

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