
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 $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci, and point $P$ is a moving point on the ellipse $E$. The perimeter of $\triangle PF\_1F\_2$ is $6$, and there exists a point $P$ such that $\triangle PF\_1F$ is a right triangle.
(1) Find the equation of the ellipse $E$;
(2) If $A$, $B$, $C$, $D$ are four distinct points on the ellipse $E$, $AC$ and $BD$ intersect at point $F\_1$, and $\overrightarrow{AC} \cdot \overrightarrow{BD} = 0$. If the slope of $AC$ is $\sqrt{3}$, find the area of the quadrilateral $ABCD$.

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