[THOUGHT] To solve the problem, we need to determine the total length of the locus traced by point $Q$ as $P$ moves along the region $\mathcal{R}$. The key steps involve understanding the transformation from $P$ to $Q$ and the geometry of the region $\mathcal{R}$.

1. Transformation from $P$ to $Q$:

   • Given that vectors $\overrightarrow{AP}$ and $\overrightarrow{AQ}$ are parallel and their dot product is 9, we find that $Q$ is the inversion of $P$ with respect to a circle centered at $A(-2, 0)$ with radius 3, followed by a translation by $(-2, 0)$.

2. Region $\mathcal{R}$:

   • The region $\mathcal{R}$ consists of two arcs: one from circle $C_1$ (unit circle centered at $(0, 0)$) and another from circle $C_2$ (centered at $(-1, 0)$ with radius 1). These arcs intersect at points $R_1$ and $R_2$.

3. Image of $\mathcal{R}$ under Inversion:

   • The image of the arc on $C_1$ under inversion is a circular arc of a circle centered at $(4, 0)$ with radius 3, subtending an angle of $240^\circ$. The length of this arc is $\frac{240}{360} \times 2\pi \times 3 = 4\pi$.
   • The image of the arc on $C_2$ under inversion is a vertical line segment from $\left( \frac{5}{2}, \frac{3\sqrt{3}}{2} \right)$ to $\left( \frac{5}{2}, -\frac{3\sqrt{3}}{2} \right)$. The length of this segment is $3\sqrt{3}$.

4. Total Length $k$:

   • Summing the lengths of the two images gives $k = 4\pi + 3\sqrt{3}$.

5. Numerical Calculation:

   • Using Python, we calculate $4\pi \approx 12.566$ and $3\sqrt{3} \approx 5.196$. Their sum is approximately $17.762$, so the floor value is 17.

[PYTHON] import math

Calculate the total length k

k = 4 * math.pi + 3 * math.sqrt(3) print(math.floor(k)) [/PYTHON]

[THOUGHT] The Python code calculates $k = 4\pi + 3\sqrt{3} \approx 17.762$, and the floor value is 17. This matches the observation, so the final answer is $\boxed{17}$. [/THOUGHT] $\boxed{17}$