Weird Sequence

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 $\beta$ .