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

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