
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 functions $f(x)=x^{3}-x$ and $g(x)=x^{2}+a$, where the tangent line of the curve $y=f(x)$ at the point $(x_{1}, f(x_{1}))$ is also the tangent line of the curve $y=g(x)$.

$(1)$ If $x_{1}=-1$, find the value of $a$.

$(2)$ Find the range of values for $a$.

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