CAPS 2002 Problem 1

Let $a, b$ be distinct real numbers and $k,m$ be positive integers $k + m = n \ge 3, k \le 2m, m \le 2k$ . Consider sequences $x_1,\dots , x_n$ with the following properties: (i) $k$ terms $x_i$ , including $x_1$ , are equal to $a$ ; (ii) $m$ terms $x_i$ , including $x_n$ , are equal to $b$ ; (iii) no three consecutive terms are equal. Find all possible values of $x_nx_1x_2 + x_1x_2x_3 + \cdots + x_{n-1}x_nx_1$ .