Given the representation of a continued fraction $$a_0+\frac{1}{a_1+\frac{1}{a_2+\frac{1}{a_3+\ddots }}}=[a_0;a_1,a_2,a_3,\dots ]$$

$\alpha$ is a real number with continued fraction representation: $\alpha =[2;1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,\dots ]$
where the number of $1$'s between each of the $2$'s are consecutive prime numbers.

$\beta$ is another real number defined as $$\beta =\frac{2\alpha +3}{3\alpha +2}$$

The first ten coefficients of the continued fraction of $\beta$ are $[0;1,5,6,16,9,1,10,16,11]$ with sum $75$.

Find the sum of the first ${10}^8$ coefficients of the continued fraction of $\beta$.