Lines & Planes in Space · Superchallenge problem

Let $L_1$ be the line through $(0, 0, 1)$ with direction $\langle 1, 1, 0\rangle$, and $L_2$ the line through $(0, 0, -1)$ with direction $\langle 1, -1, 0\rangle$. The line $M$ through the origin with direction $\langle 1, 2, 1\rangle$ contains exactly two points that are equidistant from $L_1$ and $L_2$; one of them is the origin. Find the common distance from the other point to $L_1$ and $L_2$.