
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 
$$
\begin{cases} 
x= \sqrt {2}\cos\alpha \\ 
y= \sqrt {2}+ \sqrt {2}\sin\alpha
\end{cases}
$$
($\alpha$ 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 $\theta= \frac {\pi}{4}$ intersects $C_1$ at a point $A$ different from the pole, and intersects $C_2$ at a point $B$ different from the pole, find $|AB|$.

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