Swap to Increase (Hard)

*This is the hard version of the problem. The only difference between this and the easy version are the constraints on $N$: here, $N \le 600$ and the sum of $N^3$ won't exceed $600^3$.* For a permutation$^\dagger$ $P$ of the integers $\{1, 2, \ldots, N\}$, we define $f(P)$ to be the length of the longest increasing subsequence of consecutive elements in $P$. Formally, $f(P)$ is defined to be th