
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
Given a circle with diameter $AB$, let $P$ be an arbitrary point on the circle different from $A$ and $B$. Draw maximum inscribed circles in the smaller segments above the chords $AP$ and $BP$. At what position of $P$ will the sum of the areas of the two circles be maximal, and at what position will it be minimal?

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