13 Find the maximum value of the product $x^{2} y^{2} z^{2} u$ under the conditions $x, y, z, u \geqslant 0$ and $2 x+x y+z+y z u=1$.
13. By the AM-GM inequality, we have $\sqrt[4]{2 x^{2} y^{2} z^{2} u} \leqslant \frac{2 x+x y+z+z y u}{4}=\frac{1}{4}$, which implies $x^{2} y^{2} z^{2} u \leqslant \frac{1}{512}$. Moreover, equality holds when $2 x=x y=z=y z u=\frac{1}{4}$, i.e., $x=\frac{1}{8}, y=2, z=\frac{1}{4}$, $u=\frac{1}{2}$. Therefore, the maximum value sought is $\frac{1}{512}$.