
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
Circle $F: x^{2}+y^{2}-2x=0$, parabola $C: y^{2}=4x$, the line $l$ passing through the center of the circle $F$ intersects the two curves at four points from bottom to top, denoted as $P$, $M$, $N$, $Q$ in order. If $|PM|$, $|MN|$, $|NQ|$ form an arithmetic sequence, then find the equation of line $l$.

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