Bulb Switcher II

The four flip operations commute, and pressing the same operation twice cancels it, so only each operation's parity matters. Their effects repeat with period six across bulb indices, and for the standard problem the first three bulbs distinguish all reachable states. Enumerate the at most sixteen parity combinations of the four operations, retaining only combinations whose number of active operations has the same parity as presses and is no greater than presses; extra presses can be spent in cancelling pairs. Apply each valid combination to min(n, 3) representative bulbs and insert the resulting pattern into a set. The answer is the set size. This avoids simulating n bulbs for every sequence, which would repeat equivalent states. Check presses = 0, n = 1, n = 2, and presses greater than or equal to 3, where the count stabilizes. Enumeration is O(1) time and O(1) space because there are a fixed number of operations and representative bulbs.