
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 $f(x)=(2-a)\ln x+ \frac{1}{x} +2ax(a\in\mathbb{R})$.

$(1)$ When $a=0$, find the extremum of $f(x)$;

$(2)$ When $a < 0$, find the monotonic intervals of $f(x)$;

$(3)$ When $-3 < a < -2$, if there exist $\lambda_{1}$, $\lambda_{2}\in[1,3]$ such that the inequality $|f(\lambda_{1})-f(\lambda_{2})| > (m+\ln 3)a-2\ln 3$ holds, find the range of $m$.

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