
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 $g(x)= \frac{2}{x}+\ln x$ and $f(x)=mx- \frac{m-2}{x}$, where $m\in\mathbb{R}$.

$(1)$ Find the extreme points of the function $g(x)$;

$(2)$ If $f(x)-g(x)$ is a monotonic function on $[1,+\infty)$, find the range of $m$;

$(3)$ Let $h(x)= \frac{2e}{x}$. If there exists at least one $x_0$ in $[1,e]$ such that $f(x_0)-g(x_0) > h(x_0)$ holds, find the range of $m$.

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