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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