What kind of recursion is this?

Algebra Level 5

Let $$\{a_n\}$$ and $$\{b_n\}$$ be two sequences of real numbers such that $$a_{0}b_{0} = 1$$, and, for all integers $$n \geq 0,$$ the following is satisfied:

\begin{align*} a_{n+1} &= \sqrt{6}(a_n - b_n) - \sqrt{2}(a_n + b_n) \\ b_{n+1} &= \sqrt{6}(a_n + b_n) + \sqrt{2}(a_n - b_n) . \end{align*}

Find the value of $$x,$$ to the nearest tenth, such that $$a_{2016}b_{2016} = 2^x.$$

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