TH-TMO 2011 Problem 12

$7662$ chairs are placed in a circle around the city of Chiang Mai. They are also marked with a label for either $1$ st, $2$ nd, or $3$ rd grade students, so that there are $2554$ chairs labeled with each label. The following situations happen, in order [list=i] [] $2554$ students each from the $1$ st, $2$ nd, and $3$ rd grades are given a ball as follows: $1$ st grade students receive footballs, $2$ nd grade students receive basketballs, and $3$ rd grade students receive volleyballs. [] The students go sit in chairs labeled for their grade [*] The students simultaneously send their balls to the student to their left, and this happens some positive number of times. [/list] A labelling of the chairs is called lin-ping if it is possible for all $1$ st, $2$ nd, and $3$ rd grade students to now hold volleyballs, footballs, and basketballs respectively. Compute the number of lin-ping labellings