
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 $\begin{cases} & x=-2+\dfrac{1}{2}t \\ & y=2+\dfrac{\sqrt{3}}{2}t \end{cases}$ (where $t$ is the parameter), and line $l$ intersects curve $C:{{\left( y-2 \right)}^{2}}-{{x}^{2}}=1$ at points $A$ and $B$.

$(1)$ Find the length of segment $AB$

$(2)$ With the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, establish a polar coordinate system. Suppose point $P$ has polar coordinates $\left( 2\sqrt{2},\dfrac{3\pi }{4} \right)$. 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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