
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 extreme value point on $(2,+\infty)$, find the range of values for $a$;
(II) Let $x\_1\in(0,1)$ and $x\_2\in(1,+\infty)$. If $f(x\_2)-f(x\_1)$ has a maximum value, denoted as $M(a)$, does $M(a)$ have a maximum value when $a\leqslant e+\frac{1}{e}$? If so, find the maximum value; if not, explain the reason.

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