
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.  
$(1)$ Find the equation of the ellipse;  
$(2)$ A moving line $l: mx+ny+\frac{1}{3}n=0 (m,n \in \mathbb{R})$ intersects the ellipse $C$ at points $A$ and $B$. Is there a fixed point $T$ on the coordinate plane such that the circle with diameter $AB$ always passes through point $T$? If it exists, find the coordinates of point $T$; if not, explain why.

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