
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
Consider the function $f(x)=p\left(x- \frac{1}{x}\right)-2\ln x$, where $g(x)=\frac{2e}{x}$ and $p$ is a real number.
1. If $f(x)$ has an extremum at $x=2$, find the value of $p$.
2. If $f(x)$ is monotonic on its domain, determine the range of possible values for $p$. 
3. If there exists at least one point $x_0$ in the interval $[1,e]$ such that $f(x_0) > g(x_0)$, find the range of possible values for $p$.

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