
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 XY Cartesian coordinate system, the parametric equation of line $l$ is given by:
$$
\begin{cases}
x=-2-3t \\
y=2-4t
\end{cases} \quad (t \text{ is the parameter})
$$
It intersects curve $C$ described by the equation $(y-2)^2 - x^2 = 1$ at two points $A$ and $B$.
1. Find the length of $|AB|$;
2. In a polar coordinate system with the origin $O$ as the pole and the positive x-axis as the polar axis, suppose point $P$ has polar coordinates $(2 \sqrt{2}, \frac{3\pi}{4})$. Find the distance from point $P$ to the midpoint $M$ of the line segment $AB$.

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