
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 functions $f(x)=\log _{a}x$ and $g(x)=a^{x}$, where $a \gt 1$ is a real number.  
$(1)$ When $a=e$, find the monotonic intervals and extreme values of $h(x)=\frac{x}{g(x)}$ on $(0,+\infty)$.  
$(2)$ If the equation $g(x)-\frac{f(x)}{x}=1$ has two zeros, find the range of real numbers for $a$.

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