
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 $f(x)=2x-\frac{1}{x}-a\ln x(a\in R)$.
(1) Find the monotonic intervals of $f(x)$ when $a=3$.
(2) Let $g(x)=f(x)-x+2a\ln x$, and $g(x)$ has two extreme points, one of which is $x\_1\in[0,1]$, find the minimum value of $g(x\_1)-g(x\_2)$.

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