
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 the ellipse $C_{1}: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{2}}{2}$, and the line $y = x + b$ is a tangent line of the parabola $C_{2}: y^{2} = 4x$.

(Ⅰ) Find the equation of the ellipse $C_{1}$.

(Ⅱ) A moving line $l$ passing through the point $S(0, -\frac{1}{3})$ intersects the ellipse $C_{1}$ at points $A$ and $B$. Is there a fixed point $T$ in the Cartesian coordinate plane such that the circle with diameter $AB$ always passes through $T$? If such a point exists, find the coordinates of $T$; if not, explain why.

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