
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 functions $f(x) = px - \frac{p}{x} - 2\ln x$ and $g(x) = \frac{2e}{x}$,

(Ⅰ) If $p=2$, find the equation of the tangent line to the curve $f(x)$ at the point $(1, f(1))$;  
(Ⅱ) If the function $f(x)$ is increasing throughout its domain, determine the range of the positive real number $p$;  
(Ⅲ) If $p^2-p \geq 0$, and there exists at least one point $x_0 \in [1, e]$ such that $f(x_0) > g(x_0)$, determine the range of the real number $p$.

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