
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
Let the left focus of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ be $F$, and the right vertex be $A$. Given that the eccentricity is $\frac{1}{2}$. It is known that $A$ is a focus of the parabola $y^{2}=2px\left(p \gt 0\right)$, and the distance from $F$ to the directrix $l$ of the parabola is $\frac{1}{2}$.<br/>$(Ⅰ)$ Find the equations of the ellipse and the parabola;<br/>$(Ⅱ)$ Let $P$ and $Q$ be two points symmetric about the $x$-axis on line $l$. Line $AP$ intersects the ellipse at point $B(B$ is different from $A)$, and line $BQ$ intersects the $x$-axis at point $D$. If the area of $\triangle APD$ is $\frac{\sqrt{6}}{2}$, find the equation of line $AP$.

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