
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, if the chord length intercepted by the circle $x^{2}+y^{2}=a^{2}$ on the line $x-y-\sqrt{2}=0$ is $2$
(I) Find the standard equation of the ellipse $C$;
(II) Given that points $A$ and $B$ are the two intersection points of the moving line $y=k(x-1)$, $k\neq 0$ and the ellipse $C$, does there exist a fixed point $M$ on the $x$-axis such that $\overrightarrow{MA} \cdot \overrightarrow{MB}$ is a constant value? If so, find the coordinates of point $M$ and the constant value; if not, explain the reason.

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