An alien script has $n$ letters $b_1, b_2, \dots, b_n$. For some $k < \frac{n}{2}$, assume that all words formed by any of the $k$ letters (written left to right) are meaningful. These words are called $k$-words. A $k$-word is considered **sacred** if:

1. No letter appears twice, and
2. If a letter $b_i$ appears in the word, then the letters $b_{i-1}$ and $b_{i+1}$ do not appear. (Here $b_{n+1} = b_1$ and $b_0 = b_n$).

For example, if $n = 7$ and $k = 3$, then $b_1b_3b_6$, $b_3b_1b_6$, $b_2b_4b_6$ are sacred 3-words. On the other hand, $b_1b_7b_4$, $b_2b_2b_6$ are not sacred.

What is the total number of sacred $k$-words? Use your formula to find the answer for $n = 10$ and $k = 4$.

Let's think step by step and output the final answer within \boxed{}.
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