
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 $f(x)=x-\frac{1}{x}+2a\ln x (a\in R)$,

(1) Discuss the monotonicity of $f(x)$;

(2) If $f(x)$ has two extreme values ${x_{1}},{{x_{2}}$ where ${{x_{2}}} \in \left[ {e},+{\infty} \right)$, find the minimum value of ${f}({{x_{1}})-f({{x_{2}}})$.

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