
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 eccentricity $e= \frac{\sqrt{2}}{2}$, and one of its foci coincides with the focus of the parabola $y= \frac{1}{4}x^{2}$.

(1) Find the equation of the ellipse $C$;  
(2) A moving line $l$ passing through point $S(-\frac{1}{3}, 0)$ intersects the ellipse $C$ at points $A$ and $B$. Is there a fixed point $T$ on the coordinate plane such that the circle with $AB$ as its diameter always passes through $T$ regardless of how $l$ rotates? If such a point exists, find its coordinates; if not, explain why.

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