
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 $E$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with an eccentricity $e = \frac{\sqrt{3}}{2}$, and that it passes through the fixed point $P(\sqrt{3}, \frac{1}{2})$.

$(1)$ Determine the equation of ellipse $E$;

$(2)$ Let $A$, $B$ be the left and right vertices of the ellipse $E$ respectively, and let $P$ be a moving point (not on the x-axis) on the line $l$: $x = 4$. Draw line $AP$ to intersect the ellipse at point $C$, extend line $PB$ to intersect the ellipse at point $D$. Determine whether the line $CD$ always passes through a fixed point, and if so, find this fixed point. If not, explain why.

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