
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 $f(x)=\ln x-ax-1$ $(a>0)$.  
$(1)$ When $a=0$, find the equation of the line passing through the origin and tangent to the graph of $f(x)$.  
$(2)$ If $g(x)=e^{-ax}+\frac{f(x)}{x}$ $(a>0)$ has two distinct zeros $x_{1}$ and $x_{2}$ $(0 < x_{1} < x_{2})$, and the inequality $x_{1} \cdot x_{2}^{3} > e^{m}$ always holds, find the range of real numbers for $m$.

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