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Calculus Level 4

\[\large{S = \sum_{i=1}^\infty \sum_{j=1}^\infty \sum_{k=1}^\infty \dfrac{1}{ijk(i+j+k)}}\]

If \(S\) can be expressed as \(\large \dfrac{A}{B} \pi^C \) for positive integers \(A,B,C\) with \(\gcd(A,B)=1\), submit the value of \(A+B+C\) as your answer.


Bonus: Generalize the expression below in terms of \(n\):

\[\large{P(n) = \sum_{i_1=1}^\infty \sum_{i_2=1}^\infty \cdots \sum_{i_n=1}^\infty \dfrac{1}{i_1 i_2 \dotsm i_n(i_1+i_2+\ldots+i_n)}=\ ?}\]

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