
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 ellipse $C$ has foci $F\_1(-\sqrt{2},0)$ and $F\_2(\sqrt{2},0)$, and a major axis of length $4$, let line $y = x + 2$ intersect ellipse $C$ at two distinct points $A$ and $B$.

(1) Find the equation of ellipse $C$.
(2) Find the length of chord $AB$.

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