Consider the infinite series involving the harmonic numbers \(H_n\) defined as:
\[H_n = 1 + \frac{1}{2} + \frac{1}{3} + \dots + \frac{1}{n}.\]
Evaluate the sum:
\[\sum_{n = 1}^\infty \frac{1}{n(n + 1) H_n}.\] Let's think step by step and output the final answer within \boxed{}.
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