
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
In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}{x=2+\frac{4k}{1+{k}^{2}}}\\{y=\frac{2(1-{k}^{2})}{1+{k}^{2}}}\end{array}\right.$ (where $k$ is a parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis to establish a polar coordinate system, the polar equation of curve $C_{2}$ is $\rho =\frac{2}{\sqrt{3+cos2θ-sin^{2}θ}}$. <br/>$(1)$ Write down the Cartesian equation of curve $C_{2}$ directly; <br/>$(2)$ Let $A$ be a moving point on curve $C_{1}$ and $B$ be a moving point on curve $C_{2}$, find the maximum value of $|AB|$.

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