10. Let $x, y, z$ be non-negative real numbers, and satisfy the equation $4^{\sqrt{5 x+9 y+4 z}}-68 \times 2^{\sqrt{5 x+9 y+4 z}}+256=0$, then the product of the maximum and minimum values of $x+y+z$ is $\qquad$ .
Let $m=2^{\sqrt{5 x+9 y+4 z}} \Rightarrow m^{2}-68 m+256=(m-4)(m-64)=0 \Rightarrow m=4$ or 64. When $m=4$, $\sqrt{5 x+9 y+4 z}=2 \Rightarrow 4=5 x+9 y+4 z \leqslant 9(x+y+z)$
$\Rightarrow x+y+z \geqslant \frac{4}{9}$, equality holds when $x=z=0, y=\frac{4}{9}$.
When $m=64$, $\sqrt{5 x+9 y+4 z}=6 \Rightarrow 36=5 x+9 y+4 z \geqslant 4(x+y+z)$
$\Rightarrow x+y+z \leqslant 9$, equality holds when $x=y=0, z=9$.
Therefore, the product of the maximum and minimum values of $x+y+z$ is 4.