
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 $f(x)=px- \frac{p}{x}-2\ln x$.

$(1)$ If the function $f(x)$ is increasing in its domain, find the range of values for the positive real number $p$;

$(2)$ Let the function $g(x)= \frac{2e}{x}$, if there exists at least one point ${x}_0$ in $[1,e]$ such that $f({x}_0) > g({x}_0)$ holds, find the range of values for the real number $p$.

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