Prove that: \[\sum_{k=1}^{\infty} \sum_{n=1}^{\infty} \frac{H_n}{kn (k+n)^3}=\frac{215}{48}\zeta(6) - 3\zeta^2(3)\]

**Notations:**

\(H_n\) denotes Harmonic number.

\(\zeta(\cdot) \) denotes the Riemann zeta function.

This is a part of the set Formidable Series and Integrals.

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