
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 an ellipse $E: \frac{x^2}{4} + \frac{y^2}{b^2} = 1$ with foci on the x-axis and an eccentricity of $\frac{\sqrt{6}}{3}$, suppose the vertices A, B of triangle $\triangle ABC$ are on the ellipse $E$, and vertex C is on the line $L: y = x + 2$, with $AB \parallel L$.
(1) When 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 hypotenuse AC is maximal, find the equation of the line where AB lies.

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