
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 $f(x)=a(x^{2}-x)-\ln x$ $(a\in \mathbb{R})$.
- $(1)$ If $f(x)$ has an extremum at $x=1$, find the value of $a$;
- $(2)$ If $f(x)\geqslant 0$ holds for all $x$ in $[1,+\infty)$, find the range of values for $a$.

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