Bheeshma Pitamaha and Little Abhimanyu

Abhimanyu and his great-grandfather Bheeshma are sitting together. Abhimanyu wants Bheeshma's sword as a gift. Bheeshma wants him to earn it, so he suggests a stick game. They decide to place $N$ number of sticks, each labelled with numbers from $1$ to $N$ sequentially. The game is that each player removes two sticks and places a new stick labelled with the difference of the removed stick. If Ab