
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 a moving point $P$ and two fixed points on the plane $A(-\sqrt{2},0), B(\sqrt{2},0)$, the product of the slopes of the lines connecting $P$ to $A$ and $B$ is a constant $-\dfrac{1}{2}$.

$(1)$ Try to find the equation of the trajectory $C$ of the moving point $P$;

$(2)$ Suppose the line $l: y=kx+1$ intersects the curve $C$ at points $M$ and $N$. When $\left| MN \right|=\dfrac{4\sqrt{2}}{3}$, find the equation of the line $l$.

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