Stones of the Abyss

Alice and Bob are playing a game with a single pile containing $N$ stones. They play alternating turns, and Alice always makes the first move. The rules of the game are as follows: * On her turn, Alice can remove any **odd** number of stones from the pile. * On his turn, Bob can remove any positive **even** number of stones from the pile. A player loses if they cannot make a valid move on their