
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
Let \( f(x) \) be a periodic function with a period of 2, defined on \((-\infty, +\infty)\), and an even function. On the interval \([2, 3]\), \( f(x) = -2(x-3)^2 + 4 \). Suppose the vertices \( A \) and \( B \) of a rectangle \( A B C D \) lie on the \( x \)-axis, and the vertices \( C \) and \( D \) lie on the graph of the function \( y = f(x) \) for \( 0 \leqslant x \leqslant 2 \). Find the maximum area of this rectangle.

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