
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]


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Let $n \in \mathbb{N}^*,$ and let $f(x) = \frac{\ln x}{x^n}$ and $g(x) = \frac{e^x}{x^n}$ for $x > 0$.
(1) Find the number of zeros of the function $y = f(x)$ when $n = 1$;
(2) If the graphs of the functions $y = f(x)$ and $y = g(x)$ are on opposite sides of the line $y = 1$, find the set $A$ of possible values of $n$;
(3) For all $n \in A$ and for all $x_1, x_2 \in (0, +\infty)$, find the minimum value of $|f(x_1) - g(x_2)|$.

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