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

(Ⅰ) Find the Cartesian coordinates of the intersection points of $C_2$ and $C_1$;

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

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