
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
The even function \( y = f(x) \) has a domain of \( \mathbb{R} \) and is symmetric with respect to the line \( x = 2 \). It is known that for \( x \in (-2, 2) \), \( f(x) = -x^2 + 1 \). What is the expression for \( f(x) \) in the interval \( (4k-2, 4k+2) \) where \( k \in \mathbb{Z} \)?

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