6. Let the sequence of positive numbers $x_{1}, x_{2}, \cdots, x_{n}, \cdots$ satisfy
$$
\left(8 x_{2}-7 x_{1}\right) x_{1}^{7}=8
$$

and $x_{k+1} x_{k-1}-x_{k}^{2}=\frac{x_{k-1}^{8}-x_{k}^{8}}{\left(x_{k} x_{k-1}\right)^{7}}(k \geqslant 2)$.
Find the positive real number $a$, such that when $x_{1}>a$, the sequence is monotonically decreasing, i.e., $x_{1}>x_{2}>\cdots>x_{n}>\cdots$; and when $0<x_{1}<a$, the sequence does not have monotonicity.
（Provided by Li Shenghong）