
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
In the Cartesian coordinate system $xoy$, it is known that the focal length of the ellipse $C\_1$: $\frac{y^2}{a^2} + \frac{x^2}{b^2} = 1 (a > b > 0)$ is $2\sqrt{2}$, and the point $P(0, \sqrt{3})$ is on $C\_1$.

1. Find the equation of $C\_1$;
2. Suppose the line $l$ is tangent to the ellipse $C\_1$ at point $A$ and to the parabola $C\_2$: $x^2 = 2y$ at point $B$. Find the equation of line $l$ and the length of segment $AB$.

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