
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 $f(x)=\frac{x\ln x-x+a}{x^2}$ where $a\in \mathbb{R}$.  
$(1)$ When $a=0$, find the monotonic interval of $f(x)$.  
$(2)$ If $f(x)$ has an extremum point on the interval $(1, e^{2})$, find the range of real number $a$.

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