
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
The line $l: y = x - 1$ intersects the parabola $C: y^2 = 2px$ (where $p>0$) at points $A$ and $B$, and line $l$ passes through the focus of $C$.
(1) Find the equation of the parabola $C$.
(2) If a circle $Q$ is drawn with $AB$ as its diameter, find the equation of circle $Q$.

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