
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
What is the maximum number of points \( P \) in the plane of the convex quadrilateral \( ABCD \) for which the areas of the triangles \( PAB \), \( PBC \), \( PCD \), and \( PDA \) are equal?

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