
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 the center at the origin and radius \( a \). The circle intersects the \( x \)-axis at points \( A \) and \( A' \). Let \( QQ' \) be a chord perpendicular to the diameter \( AA' \). Let \( R \) be the point where \( QQ' \) intersects the \( x \)-axis, and let \( M \) and \( M' \) be the midpoints of segments \( RQ \) and \( RQ' \), respectively. Find the locus of the intersection point \( P \) of the lines \( A'M \) and \( AM' \).

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