Let $x$ be the number of students seated around a round table. Then, let $k$ be the number of ways in which they change their seats resulting in an arrangement in which the students are either on the same seat or one of the seats adjacent to the original one.

Given that $x$ belongs to the set $(123345857,343139869)$, both not-included, find the possible number of values of $k$ which are congruent to $111 \mod 5$.