Inori Yuitsuka dreams of becoming a world-class figure skater. Despite her immense talent and passion for the sport, many people say that she is too old to begin serious training, and her mother does not want her to follow the same career path as her older sister. One day, she has a fateful meeting with Tsukasa Akeuraji, a former ice dancer, who agrees to become her coach and help her pursue her dream. Together, they work toward Inori's goal of becoming an Olympic gold medalist.

For the next competition, Inori needs to prepare a dance arrangement to perform. The ice rink is divided into $N$ equally spaced positions numbered from $1$ to $N$ in clockwise order. The distance between two adjacent positions is $1$.

Inori must perform $N$ jumps, labeled from $1$ to $N$, and each jump must be assigned to a different position on the rink. In other words, the jumps must be placed on the circle as a permutation of the $N$ positions. She will perform the jumps in the order $1, 2, 3, \dots, N$. Initially, she is at the position to perform the first jump. When she travel to a different position, she always takes the shortest path along the circle, either clockwise or counterclockwise. To match the music, the total distance she skates during the entire sequence must be exactly $M$.

As a primary school student, this task is way too complicated for Inori, so she asks for your help. Your task is to determine whether it is possible to assign the $N$ jumps to the $N$ positions so that the total travelled distance is exactly $M$. If it is possible, output any valid arrangement.