
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 $$\begin{cases} x=2+2\cos\alpha \\ y=2\sin\alpha\end{cases}$$ (where $\alpha$ is the parameter). Taking $O$ as the pole and the positive direction of the $x$-axis as the polar axis, and using the same unit length, a polar coordinate system is established.  
(Ⅰ) Write the polar equation of $C_1$;  
(Ⅱ) Suppose curve $C_2$: $\frac{x^2}{4}+y^2=1$ is transformed into curve $C_3$ through the scaling transformation $$\begin{cases} x'= \frac{1}{2}x \\ y'=y\end{cases}$$, and the ray $\theta= \frac{\pi}{3}$ ($\rho>0$) intersects $C_1$ and $C_3$ at points $A$ and $B$ respectively, find $|AB|$.

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