INMO 1992 Problem 2

If $x , y, z \in \mathbb{R}$ such that $x+y +z =4$ and $x^2 + y^2 +z^2 = 6$ , then show that each of $x, y, z$ lies in the closed interval $\left[ \dfrac{2}{3} , 2 \right]$ . Can $x$ attain the extreme value $\dfrac{2}{3}$ or $2$ ?