
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 a function $f(x)=\ln x-\frac{1}{2}ax+1$, where $a$ is a real number.  
$(1)$ If $f(x)\leqslant 0$, find the minimum value of the real number $a$.  
$(2)$ Let $g(x)=xf(x)$. If the function $g(x)$ has a maximum value and the maximum value is less than $0$, find the range of values for the real number $a$.

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