
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 an ellipse $C: \dfrac{x^{2}}{a^{2}} + y^{2} = 1$ and a circle $D: x^{2} + y^{2} = r^{2} (1 < a < r)$, points $A$ and $B$ are on the ellipse $C$, and point $E$ is on the circle $D$. The maximum and minimum values of $|AE|$ are $5$ and $1$, respectively.  
$(1)$ Find the standard equations of ellipse $C$ and circle $D$;  
$(2)$ Given that $O$ is the origin, the slopes of lines $OA$ and $OB$ are $k_{1}$ and $k_{2}$, respectively, and the slope of line $AB$ is $\dfrac{\sqrt{2}}{2}$. If quadrilateral $OAEB$ is a parallelogram, find the value of $k_{1} + k_{2}$.

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