
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 \( A B C D \) be a unit square, and let \( P \) be a point inside triangle \( A B C \). Let \( E \) and \( F \) be the points on \( A B \) and \( A D \), respectively, such that \( A E P F \) is a rectangle. Let \( x \) and \( y \) denote the lengths of \( A E \) and \( A F \), respectively.

Determine a two-variable polynomial \( f(x, y) \) with the property that the area of \( A E P F \) is greater than the area of each of the quadrilaterals \( E B C P \) and \( F P C D \) if and only if \( f(x, y)>0 \).

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