Let's determine the smallest natural number $n$ for which the inequality $\left(x^{2}+y^{2}+z^{2}\right)^{2} \leq n\left(x^{4}+y^{4}+z^{4}\right)$ holds for all real values of $x, y, z$.

---

Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.
The inequality, after squaring and rearranging, can be written in the following form:

$$
0 \leq(n-3)\left(x^{4}+y^{4}+z^{4}\right)+\left(x^{2}-y^{2}\right)^{2}+\left(y^{2}-z^{2}\right)^{2}+\left(z^{2}-x^{2}\right)^{2}
$$

On the right side, the second, third, and fourth terms are non-negative for all real numbers $x, y, z$. The first term is also non-negative if $n \geq 3$, thus the examined inequality holds identically if $n \geq 3$.

If $n<3$, then for example, by choosing $x=y=z \neq 0$, the first term on the right side is negative, and the values of the other terms are 0, so in this case, the inequality does not hold.

Thus, 3 is the smallest natural number for which the examined inequality holds for all real $x, y, z$ values.