Different counts of any type make delivery impossible. Equal counts are sufficient. Maintain the invariant that each stack is monochromatic and, when both are nonempty, their types differ. Let be the next required type.
If the second stack contains , execute 2. If the first stack contains , execute 12. Otherwise, some unread input must have type . Read input until reaching it, and deliver it immediately with 012.
Store each intervening type as follows. If the second stack has type , use 01. Otherwise, if the first stack is empty or has type , use 0. In the remaining case the second stack must be empty, so use 01. If both stacks were occupied with types different from , the three-type alphabet would force one to contain , contradicting the current case. Check the second stack's matching type first to avoid splitting one type between both stacks. Each step preserves the invariant.
Every stick traverses input, first stack, second stack, and output exactly once, so there are exactly commands. Time and space are . Subtask 1 exhaustively searches and memoizes valid states; there are at most states and time including string handling, sufficient for . For subtask 2, a single buffer stack suffices: push input until its top has the required type, then execute . With two types, the buffer is monochromatic after every delivery, giving time.
Solution written by GPT6.1
12