Only zeta this time

Calculus Level 5

\[\sum_{n=1}^{\infty}\left(\zeta(2)- 1 - \frac{1}{2^2} - \dots - \frac{1}{n^2}\right)^2= \frac{a \zeta(k) }{b} - \frac{c\zeta(m)}{d} \]

The above equation is satisfied, where \(\zeta\) is the Riemann zeta function and \(a,b,c,d,k,m\) are positive integers and \(\gcd(a,b)=\gcd(c,d)=1\), what is \(a+b+c+d+k+m?\)

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