
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 rectangular coordinate system $xOy$, the equation of line $C_1$ is $y=-\sqrt{3}x$, and the parametric equations of curve $C_2$ are given by $\begin{cases}x=-\sqrt{3}+\cos\varphi\\y=-2+\sin\varphi\end{cases}$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis.

(I) Find the polar equation of $C_1$ and the rectangular equation of $C_2$;
(II) Rotate line $C_1$ counterclockwise around the coordinate origin by an angle of $\frac{\pi}{3}$ to obtain line $C_3$, which intersects curve $C_2$ at points $A$ and $B$. Find the length $|AB|$.

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