
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ passes through the point $P(1, \frac{\sqrt{2}}{2})$, and its two foci and one endpoint of the minor axis form an isosceles right triangle.

 (Ⅰ) Find the equation of the ellipse;

 (Ⅱ) A moving line $l$ passing through the point $(0, -\frac{1}{3})$ intersects the ellipse $C$ at points $A$ and $B$. Is there a fixed point $T$ on the $y$-axis such that the circle with diameter $AB$ always passes through point $T$? 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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