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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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