For \( n ā \mathbb{N}\), there is a sequence \( a_1,a_2,a_3, \dots a_n, \dots \) such that, \( a_1=20\), \(a_2=17\) and \(\ a_n= \dfrac{2a_{n-1}a_{n-2}}{a_{n-1}+a_{n-2}} \) for all \( nā„3 \).

\[ 20^2\cdot 2017^2\cdot 17 \cdot 135576ā \sum_{k=1}^\infty \frac{1}{2017^ka_k} \]

What is the remainder when the product above is divided by \(1000\)?

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