
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 parametric equation of the curve $C_1$ is given by $\begin{cases} x=2\cos \alpha \\ y=2+2\sin \alpha \end{cases}$ ($\alpha$ is the parameter). $M$ is a moving point on $C_1$, and point $P$ satisfies $\overrightarrow{OP}=2\overrightarrow{OM}$. The trajectory of point $P$ forms the 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 of the $x$-axis as the polar axis, the ray $\theta = \frac{\pi }{3}$ intersects $C_1$ at point $A$ and $C_2$ at point $B$ (both intersections are not at the pole). Find $| AB |$.

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