
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 the circle \( P: x^{2} + y^{2} = 2x \) and the parabola \( S: y^{2} = 4x \), draw a line \( l \) passing through the center of the circle \( P \). Let the intersections of this line with the aforementioned curves, from top to bottom, be denoted as \( A, B, C, \) and \( D \). If the lengths of the segments \( AB, BC, \) and \( CD \) form an arithmetic sequence in that order, find the equation of the line \( l \).

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