
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 $g\left(x\right)=ax+a$ and $f\left(x\right)=\left\{\begin{array}{l}x^{2}-1,0\leqslant x\leqslant 2\\-x^{2},-2\leqslant x \lt 0\end{array}\right.$, determine the range of values for $a$ such that for any $x_{1}\in \left[-2,2\right]$, there exists $x_{2}\in \left[-2,2\right]$ such that $g\left(x_{1}\right)=f\left(x_{2}\right)$ holds.

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