
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ whose right focus is on the line $l$: $\sqrt{3}x - y - 3 = 0$. The product of the slopes of the lines connecting any two points symmetric about the origin on the ellipse and any other point on the ellipse is $- \frac{1}{4}$.

(1) Find the equation of the ellipse $C$.
(2) If a line $t$ passes through the point $P(1, 0)$ and intersects the ellipse $C$ at two points $A$ and $B$, does there exist 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 $\frac{d_{A}}{d_{B}} = \frac{|PA|}{|PB|}$? If so, find the value of $x_{0}$. If not, explain the reason.

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