
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 ellipse $C\_1$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ and the circle $C\_2$: $x^{2}+y^{2}=r^{2} (r > 0)$, it is known that the diameter of the circle $C\_2$ is $\sqrt{2}$ times the focal length of the ellipse $C\_1$, and the area of the circle $C\_2$ is $4π$. The eccentricity of the ellipse $C\_1$ is $\frac{\sqrt{6}}{3}$. A line $l$ with a positive slope $k$ is drawn through the upper vertex $A$ of the ellipse $C\_1$, intersecting the ellipse $C\_1$ at point $B$ and the circle $C\_2$ at points $E$ and $F$.

1. Find the equation of the ellipse $C\_1$;
2. When $|AB| \cdot |EF| = 3\sqrt{7}$, find the equation of line $l$ and the area of triangle $F\_2AB$ ($F\_2$ being the right focus of the ellipse $C\_1$).

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