
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 $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$ with eccentricity $\dfrac{\sqrt{2}}{2}$, and the area of the inscribed square of the circle with the two foci of the ellipse as its diameter is $2$.

$(1)$ Find the standard equation of the ellipse $C$;

$(2)$ If the line $l:y=kx+2$ intersects the ellipse $C$ at points $A$ and $B$, is there a point $D$ on the $y$-axis such that the sum of the slopes of lines $AD$ and $BD$, $k_{AD}+k_{BD}$, is a constant? If such a point exists, find the coordinates of point $D$ and the constant value. If not, explain why.

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