
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
For real numbers \( x \) and \( y \) within the interval \([0, 12]\):

$$
xy = (12 - x)^2 (12 - y)^2
$$

What is the maximum value of the product \( xy \)?

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