
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 \( n \) be a fixed positive integer, and let \( S \) be the set of points \((x, y)\) on the Cartesian plane such that both coordinates \( x \) and \( y \) are nonnegative integers smaller than \( 2n \) (thus \(|S| = 4n^2\)). Assume that \( \mathcal{F} \) is a set consisting of \( n^2 \) quadrilaterals such that all their vertices lie in \( S \), and each point in \( S \) is a vertex of exactly one of the quadrilaterals in \( \mathcal{F} \).

Determine the largest possible sum of areas of all \( n^2 \) quadrilaterals in \( \mathcal{F} \).

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