Eccoci.

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

$\alpha$ is a real number with continued fraction representation: $\alpha = [2;1,1,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,2,\ldots]$
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 $\beta$.