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Let the sequence of real numbers {an}n=1k\{a_n\}_{n=1}^{k} be defined as a1=3a_1 = \sqrt3, a2=1a_2 = 1, and an+2anan+1an+2=an+an+1a_{n+2} - a_na_{n+1}a_{n+2} = a_n + a_{n+1} for positive integers 1nk21 \le n \le k-2. Find the largest possible positive integer value of kk.

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