
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 \(0 < a < b\), two fixed points \(A(a, 0)\) and \(B(b, 0)\) are used to draw lines \(l\) and \(m\), respectively, which intersect the parabola \(y^2 = x\) at four different points. When these four points are concyclic, find the locus of the intersection point \(P\) of these lines \(l\) and \(m\).

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