We can substitute $u=x^2,v=y^2,w=z^2$, so that the condition becomes $7u+6v+7w=1$ with $u,v,w\geq 0$, in which case we want to find the possible values of $3u+2v+2w$. Since this is a linear function with the domain being a triangle, it must take it's minimal and maximal values on the three vertices of the triangular domain of vertices $(1/7,0,0),(0,1/6,0),(0,0,1/7)$. In this cases we get values of $3/7$, $2/6=1/3$ and $2/7$, which implies $a$ must belong to $[2/7,3/7]$.