
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 that $A_{1}$ and $A_{2}$ are the left and right vertices of the ellipse $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \; (a > b > 0)$, with $|A_{1}A_{2}| = 2\sqrt{2}$, and the eccentricity is $\frac{\sqrt{2}}{2}$.

$(1)$ Find the equation of the ellipse $E$;

$(2)$ Let the moving point $P(2\sqrt{2}, t) \; (t \neq 0)$, and denote the distances from the intersection points (different from $A_{1}$ and $A_{2}$) of lines $PA_{1}$, $PA_{2}$ with $E$ to the $x$-axis as $d_{1}$, $d_{2}$, respectively. Find the maximum value of $d_{1}d_{2}$.

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