Let’s play a game. We start with a list of numbers 1, 2, 3, . . . , 2024. We take turns by picking two numbers from the list, say a and b, and removing them. Then, we insert exactly one of the numbers |a − b| or a + b into the list. Note that the length of the list decreases by one after each change. The game stops when only one number is left on the list. Prove that the final remaining number is even.