
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 parabola $C: y^2 = 4x$, $F$ is the focus of $C$, and a line $l$ passing through point $F$ intersects $C$ at points $A$ and $B$, with $O$ being the origin.  
(1) If the slope of $l$ is 1, find the equation of the circle with $AB$ as its diameter;  
(2) If $k = 2$, find the equation of line $l$.

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