
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 Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is
\[
\begin{cases}
& x=2+t\cos \alpha \\
& y=\sqrt{3}+t\sin \alpha \\
\end{cases}
\]
where $t$ is the parameter. With the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinate equation of curve $C_2$ is $\rho =8\cos \left(\theta -\frac{\pi }{3}\right)$.

$(1)$ Find the Cartesian coordinate equations of curves $C_1$ and $C_2$, and specify what type of curves they represent;

$(2)$ If curve $C_1$ intersects with curve $C_2$ at points $A$ and $B$, find the maximum and minimum values of $|AB|$.

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