
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 the function $f(x)=ax+\ln x$, and $g(x)=a^{2}x^{2}$.  
$(1)$ When $a=-1$, find the minimum distance from the points on the graph of $y=f(x)$ to the line $x-y+3=0$;  
$(2)$ Does there exist a positive real number $a$ such that the inequality $f(x)\leqslant g(x)$ holds for all positive real numbers $x$? If it exists, find the range of $a$; if not, explain why.

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