
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 $f(x)=ax-alnx-\frac{{e}^x}{x}$.  
$(1)$ If $a=0$, find the extreme points of the function $y=f(x)$;  
$(2)$ If the inequality $f(x) \lt 0$ holds for all $x$, find the range of real numbers for $a$;  
$(3)$ If the function $y=f(x)$ has three distinct extreme points $x_{1}$, $x_{2}$, $x_{3}$, and $f({x}_{1})+f({x}_{2})+f({x}_{3})≤3{e}^{2}-e$, find the range of real numbers for $a$.

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