Question 98
Consider a regular $2022$-gon with side $1$. Its vertices are numbered $A_{1}, \ldots, A_{2022}$ in some order. Elie starts at the point $A_{1}$, then jumps from vertex to vertex to the point $A_{2}$, then from $A_{2}$ to $A_{3}$, etc., each time taking the shortest arc. When he reaches $A_{2022}$, he finally returns to $A_{1}$. Determine the maximum value of the sum of the lengths of the arcs travelled by Elie, over all possible numberings of the points.