
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 $a > 0$.

1. If $f(x)$ has an extreme value point in $(0, +\infty)$, find the range of $a$;
2. 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 $M(a)$ have a maximum value? If so, find its maximum value; if not, explain the reason.

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