Sci-fi for the New Year

Discrete Mathematics Level 3

\[\left\{\begin{array}{cccc}A \\ E \\ I \\ U\end{array}\right\} \left\{\begin{array}{cc}K \\ P\end{array}\right\} \left\{\begin{array}{cccc}A \\ E \\ I \\ U\end{array}\right\} \left\{\begin{array}{cc}R \\ S\end{array}\right\} \left\{\begin{array}{cccc}A \\ E \\ I \\ U\end{array}\right\} \left\{\begin{array}{cc}L \\ N\end{array}\right\} \left\{\begin{array}{cccc}A \\ E \\ I \\ U\end{array}\right\}.\]

A television network develops another sci-fi series, featuring a galaxy far, far away with aliens from a strange, strange planet.

Each alien has a name according to the pattern above.

Additionally, no name may contain the same vowel four times.

For instance, valid names include \(AKISANU\) and \(EPISILA\); but \(AKARALA\) is invalid.

How many different valid alien names can the series feature at most?


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