Let the sequence of real numbers $\{a_n\}_{n=1}^{k}$ be defined as $a_1 = \sqrt3$, $a_2 = 1$, and $a_{n+2} - a_na_{n+1}a_{n+2} = a_n + a_{n+1}$ for positive integers $1 \le n \le k-2$. Find the largest possible positive integer value of $k$.

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