# 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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