Weird Sequence

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Let the sequence \(\{a_n\}_{n=1}^{\beta}\) be defined as \(a_1 = \sqrt{3}\), \(a_2= 1\), and \[a_{n+2} - a_na_{n+1}a_{n+2} = a_n + a_{n+1}\] for positive integers \(1 \le n \le \alpha - 2\). Let the sequence \(\{b_n\}_{n=1}^{\beta}\) be defined as \(b_1 = -\sqrt3\), \(b_2 = 1\), and \[b_{n+2} - b_nb_{n+1}b_{n+2} = b_n + b_{n+1}\] for positive integers \(1 \le n \le \beta - 2\). Find the product of the largest possible integer values of \(\alpha\) and \(\beta\).

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