
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
A coordinate system and parametric equations are presented in this problem. In the rectangular coordinate system $xOy$, the parametric equations of curve $C_1$ are given by $\begin{cases} x=2\cos \alpha \\ y=2+2\sin \alpha \end{cases} (\alpha \text{ is the parameter})$. Point $M$ moves on curve $C_1$, and point $P$ satisfies $\overrightarrow{OP}=2\overrightarrow{OM}$. The trajectory of point $P$ forms curve $C_2$. Establish a polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis.

(1) Find the polar equation of curve $C_2$;

(2) Rotate curve $C_1$ counterclockwise around pole $O$ by $\frac{\pi}{2}$ to obtain curve $C_3$. Find the distance $|AB|$, where point $A$ is the intersection point of the ray $\theta = \frac{\pi}{3}$ and curve $C_3$ other than the pole, and point $B$ is the intersection point of the ray $\theta = \frac{\pi}{3}$ and curve $C_2$ other than the pole.

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