Given a positive sequence \(\{ a_n\} \) where \(a_1=2, a_2=5\), such that \(a_{n+1}^2=a_n \cdot a_{n+2}+3\) for all positive integers \(n\). Does there exist a constant \( b\) such that \(a_n+a_{n+2}=b \cdot a_{n+1}\) for any positive integer \(n\) ? Justify your answer.

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