
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]


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Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a>b>0$) with an eccentricity of $\frac{\sqrt{2}}{2}$, and the line $y=x+b$ is a tangent to 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 AB as its diameter always passes through T? If it exists, find the coordinates of T; if not, provide a reason.

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