Optimization & Lagrange Multipliers · Superchallenge problem

The plane $x + y + z = 0$ cuts the ellipsoid $x^2 + 2y^2 + 3z^2 = 1$ in an ellipse. Let $M$ and $m$ be the largest and smallest values of $x^2 + y^2 + z^2$ on this ellipse. Find $Mm$.