A sequence \({a_n}\) is defined as follows:

\(a_1=3, a_{k+1}=4a_k+3\)

If \(\sum_{k=1} ^{1000} \frac{1}{\log_2(a_k+1)\log_2(a_{k+1}+1)}\) can be expressed as \(\frac{m}{n}\) for positive integers \(m\) and \(n\), find \(m+n\).

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