Differentiation and Invers become one

Calculus Level 4

\[\large F(x)=f\left( e^{g(x)} \right) \]

Let \(F(x)\) be defined as above, where \( f(x)=x^3+5x+3 \) for \( 0 \leq x \leq 1\) and \( g(x)=f^{-1}(x) \). If the value of \( {F}'(3)=\dfrac{a}{b} \), where \(a\) and \(b\) are coprime positive integers, find \( a+b \).

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