
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 $xOy$, there is a curve $C_1$ defined by the parametric equations $\begin{cases} x = t \cos \alpha \\ y = t \sin \alpha \end{cases}$ (with $t$ as the parameter, $t \neq 0$), where $0 \leq \alpha < \pi$. In the polar coordinate system with $O$ as the pole and the positive x-axis as the polar axis, there are curves $C_2: \rho = 2\sin \theta$ and $C_3: \rho = 2\sqrt{3}\cos \theta$.

$(1)$ Find the Cartesian coordinates of the intersection points of $C_2$ and $C_3$.

$(2)$ If $C_1$ intersects with $C_2$ at point $A$ and with $C_3$ at point $B$, find the maximum value of the distance $|AB|$.

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