Question 128
Let $A_{1} A_{2} \ldots A_{6}$ be a regular hexagon with side length $11 \sqrt{3}$, and let $B_{1} B_{2} \ldots B_{6}$ be another regular hexagon completely inside $A_{1} A_{2} \ldots A_{6}$ such that for all $i \in \{1,2, \ldots, 5\}$, $A_{i} A_{i+1}$ is parallel to $B_{i} B_{i+1}$. Suppose that the distance between lines $A_{1} A_{2}$ and $B_{1} B_{2}$ is $7$, the distance between lines $A_{2} A_{3}$ and $B_{2} B_{3}$ is $3$, and the distance between lines $A_{3} A_{4}$ and $B_{3} B_{4}$ is $8$. Compute the side length of $B_{1} B_{2} \ldots B_{6}$.