Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$. 
Given $x,y,z\geq0$ and $x^2+y^2+z^2+x+2y+3z=13/4$. Find the minimum of $x+y+z$.

I tried many method, such as AM-GM, but all of them failed.
Thank you.

Geometrically the given equation represents a sphere and $x+y+z=t$ represents a plane which can be shifting around. So the min will be achieved on $x,y$ or $z$-axes intersecting with the sphere. Hence the possible points are $(\frac{\sqrt{14}-1}{2},0,0),(0,\frac{\sqrt{17}}{2}-1,0),(0,0,\frac{\sqrt{22}-3}{2})$, and the min is $\frac{\sqrt{22}-3}{2}$.
