
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 consider the functions $f(x)= \frac{\ln x}{x^n }$ and $g(x)= \frac{e^{x}}{x^{n}}$ for $x > 0$.

(1) When $n=1$, find the number of zeros of the function $y=f(x)$;

(2) If the graphs of the functions $y=f(x)$ and $y=g(x)$ are located on opposite sides of the line $y=1$, find the set $A$ of all possible values of $n$;

(3) For all $n \in A$ and 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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