
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 $f(x) = (a + \frac{1}{a})\ln x - x + \frac{1}{x}$, where $a > 0$,
(I) If $f(x)$ has an extreme value point on $(0, +\infty)$, find the range of $a$;
(II) Assume $a \in (1, e]$, when $x_1 \in (0, 1)$, $x_2 \in (1, +\infty)$, let the maximum value of $f(x_2) - f(x_1)$ be $M(a)$, does there exist a maximum value for $M(a)$? If so, find this maximum value, and if not, explain why.

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