
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 $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with eccentricity $\frac{\sqrt{2}}{2}$ and a length of the minor axis of $2$, the line $kx-y-k=0$ intersects the ellipse $C$ at points $A$ and $B$. <br/>$(1)$ Find the equation of the ellipse $C$; <br/>$(2)$ Does there exist a real number $k$ such that the point $M(\frac{1}{3},0)$ lies on the perpendicular bisector of the line segment $AB$? If yes, find the value of $k$; if not, explain the reason.

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