
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 point $(1, \frac{\sqrt{2}}{2})$ lies on the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and the eccentricity of the ellipse is $\frac{\sqrt{2}}{2}$.

(Ⅰ) Find the equation of the ellipse $C$;  
(Ⅱ) A line $l$ passing through the right focus $F$ of the ellipse $C$ intersects the ellipse at two points $A$ and $B$. Is there a point $M$ on the $x$-axis such that $\overrightarrow{MA} \cdot \overrightarrow{MB}$ is a constant? If it exists, find the coordinates of point $M$; if not, explain why.

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