Let $G(n)$ denote the largest possible area of an $n$-gona polygon with $n$ sides contained in the region $\{(x,y)\in {\mathbb{R}}^2:x^4\le y\le 1\}$.
For example, $G(3)=1$ and $G(5)\approx 1.477309771$.
Find $G(101)$ rounded to nine digits after the decimal point.