
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 curve $y=f(x)=a(x-1)^{2}+b\ln x$ $(a,b\in R)$ has a tangent line at the point $(1,f(1))$ with a slope of $1$.

$(1)$ If the graph of the function $y=f(x)$ is decreasing on the interval $[2,+\infty)$, find the range of values for $a$;

$(2)$ When $x\in[1,+\infty)$, the inequality $f(x)\leqslant x-1$ always holds, find the range of values for $a$.

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