
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 $\triangle ABC$ lie on the ellipse $x^{2}+3y^{2}=4$, and vertex $C$ lies on the line $l$: $y=x+2$, with $AB$ parallel to $l$.
(I) When edge $AB$ passes through the coordinate origin $O$, find the length of $AB$ and the area of $\triangle ABC$.
(II) When $\angle ABC=90^{\circ}$, and the length of the oblique edge $AC$ is at its maximum, find the equation of the line containing $AB$.

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