
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 line $y=-x+1$ and the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$, which intersect at points $A$ and $B$.

(1) If the eccentricity of the ellipse is $\frac{\sqrt{3}}{3}$ and the focal length is $2$, find the length of the line segment $AB$.

(2) If vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ are perpendicular to each other (where $O$ is the origin), find the maximum length of the major axis of the ellipse when its eccentricity $e \in [\frac{1}{2}, \frac{\sqrt{2}}{2}]$.

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