
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 $f(x) = x\ln x - \frac{1}{2}ax^{2}$, where $a \in \mathbb{R}$.  
$(Ⅰ)$ If $a=1$, find the monotonic interval of $f(x)$.  
$(Ⅱ)$ If $f(x)$ has exactly $2$ distinct extreme values, find the range of $a$.  
$(Ⅲ)$ If $f(x)$ has exactly $2$ distinct zeros, directly write down the range of $a$ (proof not required).

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