
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 two foci of the ellipse (C) are $F\_1\left(-2 \sqrt{2},0\right)$ and $F\_2\left(2 \sqrt{2},0\right)$, and the length of the major axis is $6$,

$\quad\text{(1)}$ Find the standard equation of the ellipse (C);

$\quad\text{(2)}$ Given a line passing through the point $(0,2)$ with a slope of $1$ that intersects the ellipse (C) at points A and B, find the length of the line segment AB.

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