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Calculus Level 4

Two infinite products \(A\) and \(B\) are defined as follows:

\[ A = \displaystyle\prod\limits_{n=2}^{\infty} \left(1-\frac{1}{n^3}\right), \quad B =\displaystyle\prod\limits_{n=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \]

If \(\frac{A}{B} = \frac{m}{k},\) where \(m\) and \(k\) are relatively prime positive integers, determine \(100m+k\).

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