Eleven-free Integers
An integer is called eleven-free if its decimal expansion does not contain any substring representing a power of $11$ except $1$.
For example, $2404$ and $13431$ are eleven-free, while $911$ and $4121331$ are not.
Let $E(n)$ be the $n$th positive eleven-free integer. For example, $E(3) = 3$, $E(200) = 213$ and $E(500\,000) = 531563$.
Find $E(10^{18})$.