
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 center \(O\) and diameter \(AB = 4\). Point \(C\) is the midpoint of the radius \(OB\). Construct points \(M\) and \(P\) on the circle that are symmetrical with respect to the line \(AB\) so that segment \(CP\) is perpendicular to segment \(AM\).

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