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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?