
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 that the domain of the even function $f(x)$ is $\left[-4,4\right]$, if for any $x_{1}$, $x_{2}\in \left[0,4\right]$, when $x_{1} \lt x_{2}$, it is always true that $x_{1}f(x_{1})-x_{2}f(x_{2}) \gt 0$, determine the range of values for $a$ that satisfy the inequality $\left(a+2\right)f\left(a+2\right) \lt \left(1-a\right)f\left(1-a\right)$.

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