Divisibility budget for a repeated affine block

Let $b,d,c,k\in\mathbb N$ with $\gcd(b,b+d)=1$, and $x:\mathbb N\to\mathbb N$ with $d x_0+c>0$. Assume $b x_{i+1}=(b+d)x_i+c$ for every $i<k$. Then

The positive shifted initial value bounds repetitions of a fixed affine rule. Degenerate parameters remain covered when they satisfy the stated hypotheses.