
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 $f(x)=2ax-\frac{1}{x}-(a+2)\ln x (a\geqslant 0)$.

1. When $a=0$, find the extreme value of the function $f(x)$;
2. When $a > 0$, discuss the monotonicity of the function $f(x)$;
3. When $a=1$, if for any $x\_1$, $x\_2\in[1,4]$, $|f(x\_1)-f(x\_2)| < \frac{27}{4}-2m\ln 2$ holds, find the range of the real number $m$.

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