
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 $F\_1$ and $F\_2$ are the left and right foci of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$), the ellipse $C$ passes through the point $(-\sqrt{3}, 1)$ and shares a focus with the parabola $y^2 = -8x$.

1. Find the equation of the ellipse $C$.
2. The line $l$ passing through the right focus of the ellipse $C$ with a slope of $1$ intersects the ellipse 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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