
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 line $l$ is $\begin{cases} x=-2- \frac {1}{2}t \\ y=2+ \frac { \sqrt {3}}{2}t\end{cases}$ (where $t$ is the parameter), and it intersects curve $C$: $(y-2)^{2}-x^{2}=1$ at points $A$ and $B$.  
$(1)$ Find the length of $|AB|$;  
$(2)$ With $O$ as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system. If point $P$ has polar coordinates $(2 \sqrt {2}, \frac {3\pi}{4})$, find the distance from point $P$ to the midpoint $M$ of segment $AB$.

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