
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 $C$ with its center at the origin of coordinates, its foci on the $x$-axis, the left vertex as $A$, the left focus as $F_{1}(-2,0)$, and a point $B(2,\sqrt{2})$ on the ellipse $C$. The line $y=kx \ (k\ne 0)$ intersects the ellipse $C$ at points $E$ and $F$, and the lines $AE$ and $AF$ intersect the $y$-axis at points $M$ and $N$, respectively.

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

$(2)$ Does the circle with diameter $MN$ pass through a fixed point? If it does, find the coordinates of the fixed point; if not, please explain why.

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