
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 $C$ and a hyperbola $y^{2}-x^{2}=1$ have common foci, and the eccentricity of the ellipse is $\dfrac{\sqrt{6}}{3}$.

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

$(2)$ Let $A$ be the lower vertex of the ellipse $C$, and $M$, $N$ be two different points on the ellipse other than $A$, and the product of the slopes of lines $AM$ and $AN$ is $-3$.

$\quad(1)$ Is there a fixed point that lines $M$, $N$ always pass through? If yes, find this fixed point; if not, please explain why;

$\quad(2)$ If point $P$ is on the ellipse $C$ and different from $M$, $N$, and $|MP|=|NP|$, find the minimum value of the area of $\triangle MNP$.

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