
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
Consider the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $F_1, F_2$, an eccentricity of $\frac{1}{2}$, and $P$ as its upper vertex. The area of the incircle of $\triangle F_1PF_2$ is $\frac{\pi}{3}$.

(1) Find the equation of $E$;

(2) Lines $l_1$ and $l_2$ pass through $F_1$ and $F_2$, respectively, intersecting $E$ at points $A, C$ and $B, D$, respectively, with $l_1 \perp l_2$. Find the range of possible values for the area of quadrilateral $ABCD$.

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