
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 $\begin{cases} x=2\cos \alpha, \\ y=2+2\sin \alpha \end{cases}$ ($\alpha$ is the parameter), $M$ is a moving point on $C_1$, and point $P$ satisfies $\overrightarrow{OP}=2\overrightarrow{OM}$, the trajectory of point $P$ is curve $C_2$.  
(1) Find the equation of $C_2$;  
(2) In the polar coordinate system with $O$ as the pole and the positive half-axis of $x$ as the polar axis, the ray $\theta= \dfrac{\pi}{3}$ intersects $C_1$ at point $A$ (different from the pole) and intersects $C_2$ at point $B$ (different from the pole), find $|AB|$.

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