Consider the sequence \(\{a_n\}\) given by the recurrence relation

\[\begin{cases} a_1 = 5 \\ a_2 = 8 \\ a_n + a_{n+1} = 2 a_{n+2} & \text{for } n \ge 3 \end{cases} \]

Find \(\displaystyle \lim_{n \to \infty} a_n\).

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