Via simulation and tracking the record-high positions, we find that once \(N\ge 1000\), the beaver hits a predictable groove: every \(71\) bananas, it moves exactly \(118\) steps right. \[BB(N+71)=BB(N)+118\] To fast-forward to \(10^{18}\), we just find how many \(71\) steps loops fit after our starting point of \(1000\): \[10^{18}-1000=71\times 14084507042253507 + 3\] Since there are \(3\) leftovers, the math simplifies to \[BB(10^{18})=BB(1003)+118\times 14084507042253507\]