
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
Let  $ AB$  be the diameter of a given circle with radius  $ 1$  unit, and let  $ P$  be a given point on  $ AB$ . A line through  $ P$  meets the circle at points  $ C$  and  $ D$ , so a convex quadrilateral  $ ABCD$  is formed. Find the maximum possible area of the quadrilateral.

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