Evaluate the sum of the infinite series: \[\sum_{n=1}^\infty \frac{1}{(n+1) H_n H_{n+1}},\] where \(H_n\) is the \(n\)-th harmonic number defined as \(H_n = 1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n}\). Let's think step by step and output the final answer within \boxed{}.
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