
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 that the chord length cut by the line $y=x+1$ from the circle $x^2+y^2=\frac{3}{2}$ is equal to the length of the minor axis of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2}=1$ $(a > b > 0)$, and the eccentricity of the ellipse $C$ is $e= \frac{\sqrt{2}}{2}$, find:

$(1)$ The equation of $C$.

$(2)$ Given a moving line $L$ passing through point $M(0,-\frac{1}{3})$ 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 $T$ regardless of how $L$ moves? If such a point exists, find the coordinates of $T$.

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