
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 the ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1 (a > b > 0)$ has an eccentricity of $\dfrac{\sqrt{2}}{2}$, and the point $(2,\sqrt{2})$ is on the ellipse.

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

$(2)$ Let $N(0,2)$, and a line $l$ passes through the point $P(-1,-2)$ and intersects the ellipse $C$ at two points $A$ and $B$ different from $N$, with the slopes of lines $NA$ and $NB$ being $k_1$ and $k_2$ respectively. The question is: Is $k_1+k_2$ a constant? If so, find the constant; if not, please explain why.

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