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

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