
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
Let $f(x)=ax\ln x-x+\frac{1}{2}$.  
$(1)$ If $a=1$, find the monotonic interval of $f(x)$.  
$(2)$ If for any $x\in (1,e]$, $\frac{{f(x)}}{x}+\frac{1}{{2x}}＜0$, find the range of real number $a$.

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