
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 $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ has an eccentricity of $\dfrac{1}{2}$, and the point $(\sqrt{3}, -\dfrac{\sqrt{3}}{2})$ is on the ellipse $C$.
(Ⅰ) Find the equation of the ellipse $C$;
(Ⅱ) A line $l$ passes through the right focus $F$ of the ellipse $C$ and intersects the ellipse $C$ at two distinct points $M(x_1,y_1)$ and $N(x_2,y_2)$. If point $P$ is symmetric to point $N$ with respect to the $x$-axis, determine whether the line $PM$ always passes through a fixed point. If so, find the coordinates of that point; if not, explain why.

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