Question #40475

Let $S$ be the set of all complex numbers $z$ satisfying $\text{Re}(z) = \text{Im}(z)$. For a fixed angle $\theta \in [0, 2\pi)$, a point $P$ in 3D space is defined by its coordinates $(X, Y, Z)$ where: $X = \text{Re}(z e^{i\theta}), \quad Y = \text{Im}(z e^{i\theta}), \quad Z = \text{Re}(z)$ Let $\Pi_1$ and $\Pi_2$ be two planes given by the equations: $\Pi_1: 2X + 3Y - 2Z = 1$ $\Pi_2: X + 2Y - Z = 1$ If there exists a plane $\Pi$ passing through the line of intersection of $\Pi_1$ and $\Pi_2$ such that the locus of $P$ (as the complex number $z$ varies in $S$) lies entirely in $\Pi$, find the sum of all possible values of $\theta$.