
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 $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with one of its foci at $F(\sqrt{3}, 0)$, and the distance from the major axis vertex to point $A(0, -2)$ is $2\sqrt{2}$, where $O$ is the origin.

$(1)$ Find the equation of ellipse $C$;

$(2)$ A line with a slope of $1$ passes through the right focus of ellipse $C$ and intersects the ellipse at points $A$ and $B$. Find the length of chord $AB$.

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