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

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

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

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