
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 and its eccentricity as $\frac{\sqrt{2}}{2}$, one of the foci of ellipse $C$ coincides with the focus of the parabola $x^{2}=4y$.  
$(1)$ Find the equation of ellipse $C$;  
$(2)$ A moving line $l$ passing through point $S(-\frac{1}{3}, 0)$ intersects ellipse $C$ at points $A$ and $B$. Is there a fixed point $T$ on the plane such that, no matter how line $l$ rotates, the circle with diameter $AB$ always passes through point $T$? If such a point exists, provide the coordinates of point $T$; if not, explain why.

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