Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EFGH$ be another regular tetrahedron such that the volume of $EFGH$ is $\frac{1}{8}$-th the volume of $ABCD$. The height of $EFGH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are positive coprime integers. What is $a + b$?

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