
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 $a > 0$, functions $f(x)=ax^{2}-x$, $g(x)=\ln x$.
(1) If $a=1$, find the extreme value of the function $y=f(x)-3g(x)$;
(2) Does there exist a real number $a$ such that $f(x)\geqslant g(ax)$ holds? If so, find the set of values for $a$; otherwise, explain the reason.

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