Lights Out (Hard)

*This is the hard version of the problem. Here, you must solve for every possible starting position $K$.* You're a worker on a factory floor with a row of $N$ switches. Each of them can be either ON or OFF. Initially, all switches are in the OFF state. Suppose you're standing in front of switch $i$. You can then do any one of the following two moves: - Flip switch $i$ (if it was ON, it will be