
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
We are given a convex quadrilateral  $ABCD$  in the plane.
(*i*) If there exists a point  $P$  in the plane such that the areas of  $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$  are equal, what condition must be satisfied by the quadrilateral  $ABCD$ ?
(*ii*) Find (with proof) the maximum possible number of such point  $P$  which satisfies the condition in (*i*).

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