
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 $f(x)=x^{2}-ax+2\ln x$ where $a$ is a real number.

$(1)$ Find the intervals of monotonicity for $f(x)$;

$(2)$ If $2(e+\frac{1}{e}) < a < \frac{20}{3}$, and $f(x)$ has two extreme points $x_1$ and $x_2$ ($x_1 < x_2$), find the range of $f(x_1)-f(x_2)$. (Here $e$ is the base of the natural logarithm).

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