VN-TST 1991 Problem 3

Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$ , defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$ . And for $n \geq 1$ we have:

$$x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.$$

Show that this sequence has a finite limit. Determine this limit.