
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 $f(x)=a\ln x-x^{2}+1$.

$(1)$ If the tangent line of the curve $y=f(x)$ at $x=1$ is $f(x)=a\ln x-x^{2}+1$, find the real number $a$ and the value at $x=1$;

$(2)$ Discuss the monotonicity of the function $4x-y+b=0$;

$(3)$ If $a < 0$, and for any $x_{1}, x_{2} \in (0, +\infty)$, it holds that $|f(x_{1})-f(x_{2})| \geqslant |x_{1}-x_{2}|$, find the range of values for $a$.

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