Nik wants to meet Anya, and so he fires up his teleporter. The teleporter works by independently choosing a random real number between $0$ and $1$ thrice. Suppose the three numbers it chooses are $X, Y$, and $Z$. The teleporter then transports Nik to $x = \max(X, Y, Z)$.

As Anya is smart, she will wait for Nik at the coordinate where the expected distance from her to where Nik lands will be the minimum. If she waits at $x = L$, find $\lfloor L\cdot10^9\rfloor$, where $\lfloor x \rfloor$ denotes the integer part of $x$.