Fermat to infinity

Algebra Level 5

$\left(1-\frac{1}{a_1}\right)\left(1-\frac{1}{a_2}\right)\left(1-\frac{1}{a_3}\right)\left(1-\frac{1}{a_4}\right) \cdots$

Let $$a_1=\frac{17}{4}$$ and $$a_n=a_{n-1}^2-2$$ for $$n \geq 2$$.

If the infinite product above can be expressed as $$\frac AB$$, where $$A$$ and $$B$$ are coprime positive integers, then what is the value of $$A+B$$?

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