
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 $\alpha \in (0, \pi)$. Let M be a moving point on $C_1$, satisfying the condition $$\overrightarrow{OM} = 2\overrightarrow{OP}$$, where P is the trajectory of curve $C_2$.  
(Ⅰ) Find the standard equation of $C_2$;  
(Ⅱ) In the polar coordinate system with O as the pole and the non-negative half-axis of x as the polar axis, the ray $$\theta= \frac{\pi}{3}$$ intersects $C_1$ and $C_2$ at points A and B, respectively. Calculate $|AB|$.

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