
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 $f(x) = ax - \frac{a}{x} - 2\ln x$ $(a > 0)$.

(Ⅰ) When $a=2$, find the number of zeros of the function $f(x)$;

(Ⅱ) Discuss the monotonicity of $f(x)$;

(Ⅲ) Let the function $g(x) = \frac{2e}{x}$, if there is at least one point $x_0$ in the interval $[1, e]$ such that $f(x_0) > g(x_0)$ holds, find the range of the real number $a$.

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