CGMO 2008 Problem 6

Let $(x_1,x_2,\cdots)$ be a sequence of positive numbers such that $(8x_2 - 7x_1)x_1^7 = 8$ and $$x_{k + 1}x_{k - 1} - x_k^2 = \frac {x_{k - 1}^8 - x_k^8}{x_k^7x_{k - 1}^7} \text{ for }k = 2,3,\ldots$$

Determine real number $a$ such that if $x_1 > a$ , then the sequence is monotonically decreasing, and if $0 < x_1 < a$ , then the sequence is not monotonic.