
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 $f(x) = \frac{x}{\ln x}$, and $g(x) = f(x) - mx (m \in \mathbb{R})$,

(I) Find the interval of monotonic decrease for function $f(x)$.

(II) If function $g(x)$ is monotonically decreasing on the interval $(1, +\infty)$, find the range of the real number $m$.

(III) If there exist $x_1, x_2 \in [e, e^2]$ such that $m \geq g(x_1) - g'(x_2)$ holds true, find the minimum value of the real number $m$.

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