Monotonous

Calculus Level 4

Define \(\displaystyle f(x) = \ln x, F_n(f(x)) =\prod_{r=1}^n f^{(r)} (x) \) for positive integers \(n\).

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 \(f(x)\).

This is an original problem.
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