4. Let $x, y, z$ be non-negative real numbers, and satisfy the equation
$$
4^{\sqrt{5 x+9 y+4 z}}-68 \times 2^{\sqrt{5 x+9 y+4 x}}+256=0 \text {, }
$$

Then the product of the maximum and minimum values of $x+y+z$ is $\qquad$ .
4. Let $x=2^{\sqrt{5 x+9 y+4 z}}$, then
$$
x^{2}-68 x+256=0 \text {. }
$$

Solving the equation, we get $x=4,64$.
When $5 x+9 y+4 z=4$, from the inequality
$$
4(x+y+z) \leqslant 5 x+9 y+4 z \leqslant 9(x+y+z),
$$

we know
$$
4(x+y+z) \leqslant 4, \quad 9(x+y+z) \geqslant 4 .
$$

Thus, at this time, the minimum value of $x+y+z$ is $\frac{4}{9}$, and the maximum value is 1.
Similarly, when $5 x+9 y+4 z=36$, the minimum value of $x+y+z$ is 4, and the maximum value is 9.
Therefore, the minimum value of $x+y+z$ is $\frac{4}{9}$, and the maximum value is 9, and their product is 4.