
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 $f(x) = a\ln x - x + \frac{1}{x}$, where $a > 0$:
(I) If $f(x)$ has an extremum in the interval $(2, +\infty)$, find the range of $a$.
(II) Let $x_1 \in (0, 1)$ and $x_2 \in (1, +\infty)$. If there is a maximum value for $f(x_2) - f(x_1)$, denoted as $M(a)$, determine whether there is a maximum value of $M(a)$ when $a \leq e + \frac{1}{e}$. If it exists, find the maximum value; if not, provide the reason.

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