Problem Analysis #29

Consider the point $A(-2,0)$ in the coordinate plane. Suppose two points $P$ and $Q$ satisfy the following conditions:

• [(i)] The vectors $\overrightarrow{AP}$ and $\overrightarrow{AQ}$ are parallel.
• [(ii)] The dot product satisfies $\overrightarrow{AP} \cdot \overrightarrow{AQ} = 9$. Let the two circles$C_{1} : x^{2} + y^{2} = 1, \qquadC_{2} : (x+1)^{2} + y^{2} = 1$intersect at two distinct points $R_{1}$ and $R_{2}$.Consider the region $\mathcal{R}$ consisting of the arc $R_{1}O'R_{2}$ of circle $C_{1}$,which contains the point $O'(-1,0)$, together with the arc $R_{1}OR_{2}$ of circle $C_{2}$,which contains the point $O(0,0)$.When the point $P$ moves along the region $\mathcal{R}$, let $k$ denote the total lengthof the locus traced by the corresponding point $Q$.For any real number $s$, let $\lfloor s \rfloor$ denote the greatest integer less than or equal to $s$.Determine the value of $\lfloor k \rfloor$.