
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 $xoy$, line $l$ passes through point $P_0(-2,2)$, and its angle of inclination $\alpha =\dfrac{\pi }{6}$,

Line $l$ intersects the circle: ${{x}^{2}}+{{(y-2)}^{2}}=2$ at points $A$ and $B$.

$(1)$ Write the parametric equation of line $l$, and calculate the length of segment $AB$;

$(2)$ With the origin $O$ as the pole and the positive $x$-axis as the polar axis, the polar coordinates of point $P$ are $\left(2 \sqrt{2}, \dfrac{3}{4}\pi\right)$. Let the midpoint of $AB$ be $Q$, calculate the distance between points $P$ and $Q$.

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