Question 162
A *state* refers to a way to place the numbers $1, 2, \ldots, 99$ on the vertices of a given regular $99$-gon, with exactly one number on each vertex and every number appearing exactly once. Two states are considered *equivalent* if one of them can be obtained from the other by rotating the regular $99$-gon (on the plane).
An *operation* is to first choose two adjacent vertices of the regular $99$-gon and then exchange the numbers on these two vertices. Determine the smallest positive integer $N$ such that, for any two states $\alpha$ and $\beta$, one may apply no more than $N$ operations to $\alpha$ to obtain a state that is equivalent to $\beta$.
An *operation* is to first choose two adjacent vertices of the regular $99$-gon and then exchange the numbers on these two vertices. Determine the smallest positive integer $N$ such that, for any two states $\alpha$ and $\beta$, one may apply no more than $N$ operations to $\alpha$ to obtain a state that is equivalent to $\beta$.