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

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

(II) When $a < 0$, determine the intervals on which $f(x)$ is monotonic;

(III) If for any $a \in (-3,-2)$ and $x_1, x_2 \in [1,3]$, the inequality $(m + \ln 3)a - 2\ln 3 > |f(x_1) - f(x_2)|$ always holds, find the range of values for the real number $m$.

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