ELMO SL 2014 N10

Find all positive integer bases $b \ge 9$ so that the number $$\frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3}$$ is a perfect cube in base 10 for all sufficiently large positive integers $n$ .

Proposed by Yang Liu