
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 the hyperbola $C\_1$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ passes through point $P(\sqrt{2}, \sqrt{2})$ and has an eccentricity of $\sqrt{3}$.

1. Find the equation of $C\_1$;
2. An ellipse $C\_2$ passes through point $P$ and shares the same foci with $C\_1$. Line $l$ passes through the right focus of $C\_2$ and intersects $C\_2$ at points $A$ and $B$. If a circle passes through point $P$ and has line segment $AB$ as its diameter, find the equation of line $l$.

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