Sure, here is the translated text:

```
II. (40 points) Find all positive integers $n$, such that for any positive real numbers $a, b, c$ satisfying $a+b+c=1$, we have
$$
a b c\left(a^{n}+b^{n}+c^{n}\right) \leqslant \frac{1}{3^{n+2}}.
$$
```