In the language of the mumbo-jumbo tribe, there are only 4 letters: $a, u, o$ and $e$. The letter $e$ is special: if we insert $e$ into a word - once or multiple times, at the beginning, end, or somewhere in between - the meaning of the word does not change. Additionally, the letter groups aаaaaaa, ииuиuии, ооооооo in a word can be replaced by 1 $e$ without changing the meaning of the word. The meaning of a word also does not change if the aaau group in it is replaced by ua, the oou group by $o u$, or the aaaao group by oa.

The tribe has 400 members. Could each of them have a different name?
Let's consider an arbitrary word. We will show that its meaning always coincides with the meaning of a word of the form

$$
\underbrace{a a \ldots a o}_{p \text{ times}} \underbrace{o o \ldots o}_{q \text{ times}} \underbrace{u u u \ldots u}_{r \text{ times}} \quad(0 \leq p, q, r \leq 6)
$$

The $e$ letters in the word can be omitted. If there is a $u$ letter followed by an $o$ or $a$ to its right, then the $u o$ or $u a$ group can be replaced by $o u$ or $aaaau$, respectively. This way, we can achieve - without changing the meaning of the word - that only $u$ letters stand at the end of the word. If more than 6 $u$ letters stand at the end of the word, we can replace the uuuuuuu group with $e$, and then omit $e$, to ensure that fewer than 7 $u$ letters stand at the end of the word.

The $oa$ letter pair can be replaced with the $aaaao$ letter group, so we can achieve that only $a$ letters stand at the beginning of the word, and only $o$ letters stand in the middle of the word. By the already mentioned method, we can ensure that neither the number of $o$s nor the number of $a$s is too large, meaning that the original word has the same meaning as a word of form (1).

We only need to examine how many words of form (1) there can be. Since at most 7 of each letter can appear, the number of words of form (1) is $7^3 = 343$. Since $400 > 343$, it is not possible for each member of the club to have a different name.