
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  $\Omega$  be a unit circle and  $A$  be a point on  $\Omega$ . An angle  $0 < \theta < 180^\circ$  is chosen uniformly at random, and  $\Omega$  is rotated  $\theta$  degrees clockwise about  $A$ . What is the expected area swept by this rotation?

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