
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 functions $f(x)$ and $f(x + 2)$ are both even functions defined on the real numbers $\mathbb{R}$, and for $x \in [-2, 2]$, $f(x) = g(x)$, determine the analytical expression of $f(x)$ when $x \in [-4n - 2, -4n + 2]$ where $n \in \mathbb{Z}$.

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