
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 \( 0 < a < b \), lines \( l \) and \( m \) pass through two fixed points \( A(a, 0) \) and \( B(b, 0) \), respectively, intersecting the parabola \( y^2 = x \) at four distinct points. When these four points lie on a common circle, determine the locus of the intersection point \( P \) of lines \( l \) and \( m \).

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