
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
The parametric equation of curve $C_1$ is $\begin{cases} & x=2\cos \alpha \\ & y=2+2\sin \alpha \\ \end{cases}$ (where $\alpha$ is the parameter), $M$ is a moving point on curve $C_1$, and $M$ is the midpoint of segment $OP$, the trajectory of point $P$ is curve $C_2$, the polar equation of line $l$ is $\rho \sin \left( \theta +\frac{\pi }{4} \right)=\sqrt{2}$, line $l$ intersects curve $C_2$ at points $A$ and $B$.

$(1)$ Find the standard equation of curve $C_2$; $(2)$ Find the length of segment $AB$.

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