
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 centered at the origin with radius \(a\), let the points of intersection of the circle with the \(x\)-axis be \(A\) and \(A'\). Consider a chord \(QQ'\) 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 \(RQ\) 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}
user