
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
Let ellipse $E$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ have its right focus at a distance of $3$ from the line $x - y + 2\sqrt{2} = 0$, and it passes through the point $(-1, -\dfrac{\sqrt{6}}{2})$.

$(1)$ Find the equation of $E$;  
$(2)$ Let the left vertex of ellipse $E$ be $A$, and the line $l$: $x - my - t = 0$ intersects the ellipse $E$ at two distinct points $M$, $N$ (both $M$ and $N$ are different from $A$), and the circle with diameter $MN$ passes through point $A$. Determine whether line $l$ passes through a fixed point, and if so, find the coordinates of that fixed point.

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