
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 $E$: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 \quad (a>b>0)$$ with its left and right foci respectively, and point $P$ is a moving point on ellipse $E$. The perimeter of $\triangle PF_{1}F_{2}$ is 6, and there exists a point $P$ such that $\triangle PF_{1}F_{2}$ is an equilateral triangle.
1. Find the equation of ellipse $E$;
2. If $A$, $B$, $C$, $D$ are four distinct points on 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 quadrilateral $ABCD$.

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