# 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 $$f(x)$$.

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