
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 a line $y=-x+1$ and an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), they intersect at points A and B.
(1) If the eccentricity of the ellipse is $\frac{\sqrt{3}}{3}$ and the focal distance is 2, find the length of the segment AB.
(2) (For Liberal Arts students) If segment OA is perpendicular to segment OB (where O is the origin), find the value of $\frac{1}{a^2} + \frac{1}{b^2}$.
(3) (For Science students) If segment OA is perpendicular to segment OB (where O is the origin), and when the eccentricity of the ellipse $e$ lies in the interval $\left[ \frac{1}{2}, \frac{\sqrt{2}}{2} \right]$, find the maximum length of the major axis of the ellipse.

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