Residue of a sequence

Consider a sequence \(\{a_i\}\) of positive integers defined by \(a_1= 1, a_2= 2\), and for all integers \(n>2\), \[a_n= 3a_{n-1} + 5a_{n-2} \] Consider the set \[S= \{a_1, a_2, \cdots , a_{1200} \} \] Sam randomly picks an element from this set. The probability that this element is a multiple of \(8\) can be expressed as \(\dfrac{a}{b}\), where \(a, b\) are coprime positive integers. Find \(a+b\).

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