
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 $f(x)=\left(a+\frac{1}{a}\right)\ln x-x+\frac{1}{x}$, where $a > 0$.

(I) If $f(x)$ has an extreme value point in $(0,+\infty)$, find the range of $a$;

(II) Suppose $a\in(1,e]$, when $x_{1}\in(0,1)$, $x_{2}\in(1,+\infty)$, denote the maximum value of $f(x_{2})-f(x_{1})$ as $M(a)$. Does $M(a)$ exist a maximum value? If it exists, find its maximum value; If not, explain the reason.

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