AMO-Bench 2025 Problem 35

In a game, a positive integer $n$ not exceeding 10000 is initially written on a blackboard. Two players, Player 1 and Player 2, take turns making moves, with Player 1 going first. A move consists of erasing a positive integer $k$ greater than 1 from the blackboard and replacing it with five non-negative integers that sum to $k$ and whose pairwise differences are at most 1. For example, the integer 11 could be replaced with 2, 2, 2, 2, 3. We also stipulate that 1 couldn't be replaced with five integers. The game ends when a player cannot make a valid move, at which point the other player wins. Assuming both players use an optimal strategy and Player 1 wins, how many possible values of $n$ are there?