
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
Square \( ABCD \) has a side length of 40. Point \( P \) is located inside the square such that \( AP = 16 \) and \( BP = 30 \). Calculate the area of the quadrilateral formed by the centroids of triangles \( \triangle ABP \), \( \triangle BCP \), \( \triangle CDP \), and \( \triangle DAP \).

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