
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 $C$ with foci at $F_{1}(-2 \sqrt {2}, 0)$ and $F_{2}(2 \sqrt {2}, 0)$, and the major axis is 6.  
(1) Find the standard equation of ellipse $C$;  
(2) A line passing through the point $(0, 2)$ with a slope of 1 intersects the ellipse $C$ at points $A$ and $B$. Find the length of segment $AB$.

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