
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 $f(x)=\frac{x}{\ln x}-ax$ where $a\geq 0$.  
$(1)$ If $a=0$, find the intervals where the function $f(x)$ is monotonic.  
$(2)$ If there exists $x_1 \in [e, e^2]$ such that $f(x_1) \leq \frac{1}{4}$, determine the range of real numbers for $a$.

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