Define \(\displaystyle f(x) = \ln x, F_n(f(x)) =\prod_{r=1}^n f^{(r)} (x) \) for positive integers \(n\).
###### This is an original problem.

And denote \( \displaystyle G_n(x) = \sum_{i=1}^n f \left(x^i \right ) \) such that \( S(n) = F_n(G_n(1)) \).

What is the value of \( |4S(3)| + S(1) \)?

**Details and Assumptions**:

\(f^{(r)} (x) \) denote the \(r\)-th derivative of \(f(x)\).

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