
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 that the vertices A and B of $\triangle ABC$ are on the circle $x^2+y^2=4$, and C is on the line $l: y=x+2$, with $AB \parallel l$.  
(1) When the side AB passes through the origin O, find the length of AB and the area of $\triangle ABC$;  
(2) When $\angle ABC=90^\circ$, and the length of the hypotenuse AC is maximized, find the equation of the line where AB lies.

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