A product sum inequality question

For any $x_{1},x_{2},\cdots x_{6}$ with $\sum_{i=1}^{6}x_{i}^{2}=1$ and $y_{1},y_{2},\cdots y_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}y_{i}^{2}=1$, do there always exist $z_{1},z_{2},\cdots z_{6}$ in $\mathbb{R}$ with $\sum_{i=1}^{6}z_{i}^{2}=6$ such that $\left|z_{1}z_{2}z_{3}z_{4}\sum_{i=1}^{6}x_{i}z_{i}\sum_{j=1}^{6}y_{j}z_{j}\right|\ge1$?

For special cases such as $x_iy_i\ge0$, one can see it holds. But for the general case, I am stuck. Could anyone help on this question?

Maybe I'm misreading, but it looks like setting $y_1=y_2=y_3=y_4=0$ gives an easy counterexample. –  Brendan McKay Dec 12 '11 at 9:02