Xor Minimization (Hard)

**This is the hard version of the problem. In this version, you must find the lexicographically maximum permutation that satisfies the conditions. This problem is worth $40$ points in divisions $1$ and $2$, and is not scorable in divisions $3$ and $4$.** You are given an integer $N$ ($N\geq 3$). Find a permutation$^\dagger$ $P$ of the integers $1$ to $N$ such that $P_i \neq i$ for every index