On certain rational recursive sequences of order four

We obtain solutions to the recursive sequences of the form $$x_{n+1}=\frac{x_{n-3}{x}_n}{x_{n-2}(a_n+b_n{x}_{n-3}{x}_n)}$$ where $a_n$ and $b_n$ are arbitrary sequences of real numbers, and the initial values are gives as; $x_{-3},x_{-2},x_{-1}$ and $x_0$. Our methodology is to employ a group theoretic method which lowers the order of the equations, and then solve the resulting lower order recurrence relations that arise therefrom.