
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\frac{\sqrt{3}}{2}$, and the point $(\sqrt{2}, \frac{\sqrt{2}}{2})$ is on $C$.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) A line $l$ passes through the point $P(1,0)$ and intersects the ellipse $C$ at two points $A$ and $B$. Is there a line $l_0$: $x=x_0$ (where $x_0 > 2$) such that the distances $d_A$ and $d_B$ from $A$ and $B$ to $l_0$ satisfy $\frac{d_A}{d_B} = \frac{|PA|}{|PB|}$ always holds? If it exists, find the value of $x_0$; if not, explain why.

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