A bee is moving in three-dimensional space. A fair six-sided die with faces labeled $A^{+}$, $A^{-}$, $B^{+}$, $B^{-}$, $C^{+}$, and $C^{-}$ is rolled. Suppose the bee occupies the point $(a,b,c)$. If the die shows $A^{+}$, then the bee moves to the point $(a+1,b,c)$ and if the die shows $A^{-}$, then the bee moves to the point $(a-1,b,c)$. Analogous moves are made with the other four outcomes. Suppose the bee starts at the point $(0,0,0)$ and the die is rolled four times. What is the probability that the bee traverses four distinct edges of some unit cube, such that its final position is not $(0,0,0)$?