
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 point $A$ is the left vertex of the ellipse $C: \frac{x^2}{9} + \frac{y^2}{t} = 1 (t > 0)$, and the line $l: x=my+1 (m \in \mathbb{R})$ intersects the ellipse $C$ at points $E$ and $F$, and intersects the $x$-axis at point $B$. When $m=0$, the area of $\triangle AEF$ is $\frac{16}{3}$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Suppose the lines $AE$ and $AF$ intersect the line $x=3$ at points $M$ and $N$ respectively, determine whether the circle with diameter $MN$ passes through point $B$, and explain the reason.

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