
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 equations of line $l$ are $\begin{cases} x=-2-3t, \\ y=2-4t. \end{cases} (t \text{ is the parameter})$, and it intersects with 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 established with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinates of point $P$ are $(2 \sqrt{2}, \frac{3π}{4})$. Find the distance between point $P$ and the midpoint $M$ of segment $AB$.

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