If

\[\sum_{n=1}^\infty \frac{H_n^{(2)}}{n^2}\]

can be expressed in the form \(\dfrac{a\pi^k}{b}\), where \( a, b\), and \(k\) are positive integers with \(a\) and \(b\) coprime, find \(a+b+k\).

**Notation**: \(\displaystyle H_n^{(s)} = \sum_{m=1}^n \frac{1}{m^s}\).

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