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

(I) Find the Cartesian coordinates of the intersection points of $C_2$ and $C_3$;

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

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