
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 $f(x) = \frac{x(1-a\ln x)}{\ln x}$.

(1) If $a=0$, find the equation of the tangent line at point $P(e, f(e))$.

(2) If the function $f(x)$ is monotonically decreasing on the interval $(1, +\infty)$, find the minimum value of the real number $a$.

(3) If there exist $x_1, x_2 \in [e, e^2]$ such that $f(x_1) - a \leq f'(x_2)$, find the range of the real number $a$.

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