
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 $C: \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ where $a > b > 0$, with eccentricity $\dfrac{1}{2}$. A line $l$ passing through the right focus $F$ of the ellipse intersects the ellipse at points $A$ and $B$. When the line $l$ is perpendicular to the $x$-axis, $|AB| = 3$.
- (Ⅰ) Find the standard equation of the ellipse $C$.
- (Ⅱ) If the line $l$ is not perpendicular to the $x$-axis, does there exist a point $P$ on the $x$-axis (different from point $F$) such that the distances from any point on the $x$-axis to the lines $PA$ and $PB$ are equal? If yes, find the coordinates of point $P$; if not, explain why.

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