
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
Elective 4-4: Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, there is a line $l: \begin{cases}x=t \\ y= \sqrt{5}+2t\end{cases} (t$ is the parameter$)$, with the origin $O$ as the pole and the positive $x$-axis as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is ${\rho}^{2}\cos 2\theta+4=0$.

$(1)$ Write the Cartesian equation of curve $C$;

$(2)$ Given point $A(0, \sqrt{5})$, line $l$ intersects curve $C$ at points $M$ and $N$, find the value of $\dfrac{1}{|AM|}+ \dfrac{1}{|AN|}$.

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