confusing lagrange multiplier question

f(x,y,z)=xy+z^3

subject to x^2+y^2+z^2=1

Find all the absolute maximum and minimum.

I tried to solve the simutaneous equations, and I ended up with 14 critical points! (Headbang)

So I doubt if I did right or wrong.

Can anybody please help me and see how many of them? What are they? I solved that x=o, plus or minus square root 2 on 2, plus or minus 2/3

y=0, plus or minus square root 2 on 2, plus or minus 2/3

z=0, plus or minus 1, plus or minus 1/3

Which rearrange them, I end up with 14 critical points. Am I right? Thanks a lot.