
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 eccentricity $\frac{\sqrt{2}}{2}$, and the area of the quadrilateral formed by the four vertices of the ellipse is $2\sqrt{2}$.  
$(1)$ Find the equation of $E$;  
$(2)$ Through the left focus $F_1$ of $E$, draw line $l_1$ intersecting $E$ at points $A$ and $B$, and through the right focus $F_2$, draw line $l_2$ intersecting $E$ at points $C$ and $D$, with $l_1 \parallel l_2$. The area of the quadrilateral formed by points $A$, $B$, $C$, and $D$ is $S = \frac{8}{3}$. Find the equations of $l_1$ and $l_2$.

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