Question #40869

Let $A(-1, 0, 2)$ and $B(1, 0, 1)$ be two points in $\mathbb{R}^3$. A transformation $T: \mathbb{R}^3 \to \mathbb{C}$ maps a point $(x,y,z)$ to the complex plane via $T(x,y,z) = z + (x+iy)e^{i\pi/2}$. Let $L$ be the line passing through $A$ and $B$. There exist exactly three distinct points $P_1, P_2, P_3$ on $L$ such that for each point $P_j$, its image $w_j = T(P_j)$ satisfies the condition that $w_j^3$ is a purely real number. If $P_j$ divides the directed line segment $\vec{AB}$ in the ratio $k_j:1$ (i.e., $\vec{AP_j} = \frac{k_j}{k_j+1} \vec{AB}$ where $k_j \in \mathbb{R}$), find the value of $\left(\sum_{j=1}^3 k_j\right)^2 + 2 \prod_{j=1}^3 k_j$.