In coordinate space, let $O$ be the origin and $E$ be the plane $x - z = 4$.
It is known that there is a point $P(a, b, c)$ in space such that the angle $\theta$ between vector $\overrightarrow{OP}$ and vector $(1, 0, 0)$ satisfies $\theta \leq \frac{\pi}{6}$. Show that the real numbers $a, b, c$ satisfy the inequality $a^{2} \geq 3\left(b^{2} + c^{2}\right)$. (Non-multiple choice question, 4 points)