
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
Rectangle $ABCD$ is inscribed in the region bound by the parabola \( y = x^2 - 12x + 32 \) and the $x$-axis. The vertex $A$ of the rectangle lies on the $x$-axis, and the rectangle's two sides are parallel and two sides are perpendicular to the $x$-axis. Each vertical side of the rectangle has a length equal to the third of the rectangle's base. Find the area of the rectangle $ABCD$.

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