A calculus problem by Daniel Thompson

Calculus Level pending

Let \(F\) be defined by \[F(x)=\int_{0}^{x^3}\left (\int_{0}^{y^2} \left (\int_{0}^{z} x^3y^2zt \; \mathrm{d}t \right )\mathrm{d}z \right )\mathrm{d}y\]

\(F' \left (\sqrt[7]{2}\right)=\frac{a}{b}\) for relatively prime positive integers \(a\) and \(b\). What is the value of \(a+b\)?

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