
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 $f(x)=x-\frac{1}{x}-a\ln x (a\in R)$.
(1) Find the monotonic intervals of $f(x)$;
(2) Let $g(x)=f(x)+2a\ln x$, and $g(x)$ has two extreme points $x\_1$, $x\_2$, where $x\_1\in(0,e]$, find the minimum value of $g(x\_1)-g(x\_2)$.

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