
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 the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $\frac{\sqrt{2}}{2}$, and the chord length intercepted by the line $x - y - \sqrt{2} = 0$ on the circle $x^2 + y^2 = a^2$ 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 $y = k(x - 1)$, $k \neq 0$, with the ellipse $C$, is there a fixed point $M$ on the x-axis such that $\overrightarrow{MA} \cdot \overrightarrow{MB}$ is a constant? If it exists, find the coordinates of point $M$ and the constant value; if not, explain why.

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