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

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