
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
In a plane, there is a convex quadrilateral \(ABCD\).
(1) If there exists a point \(P\) in the plane such that the areas of triangles \(\triangle ABP\), \(\triangle BCP\), \(\triangle CDP\), and \(\triangle DAP\) are all equal, what conditions must the quadrilateral \(ABCD\) satisfy?
(2) What is the maximum number of such points \(P\) that can exist in the plane? Provide a proof for your conclusion.

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