The triangle $\mathrm{△}ABC$ is inscribed in an ellipse with equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $0<2b<a$, $a$ and $b$ integers.

Let $r(a,b)$ be the radius of the incircle of $\mathrm{△}ABC$ when the incircle has center $(2b,0)$ and $A$ has coordinates $(\frac{a}{2},\frac{\sqrt{3}}{2}b)$.

For example, $r(3,1)=\frac{1}{2}$, $r(6,2)=1$, $r(12,3)=2$.

Let $G(n)=\sum _{a=3}^n\sum _{b=1}^{\lfloor \frac{a-1}{2}\rfloor }r(a,b)$

You are given $G(10)=20.59722222$, $G(100)=19223.60980$ (rounded to $10$ significant digits).

Find $G({10}^{11})$.

Give your answer in scientific notation rounded to $10$ significant digits. Use a lowercase e to separate mantissa and exponent.

For $G(10)$ the answer would have been 2.059722222e1.