I was doing my homework and got really confused about how to approach the optimal levels of inputs when there are three variables. My current understanding is that the problem is to solve the optimization problem
$$\min_{H, L, K}\;{sH + wL +rK}$$ subject to $$q = \min\{H,L\} + \min\{H,K\}$$
However, I don't know how to determine the inputs of H, L, and K. Below is the problem:
A monopolist can hire high skill labor $H$, low skill labor $L$, and robots $K$. The per-unit price of these inputs are $s$, $w$, and $r$. High skill labor is more expensive: $s > w$. The production function is $f(H, L, K) = \min\{H, L\} + \min\{H, K\}$. The demand curve that the monopolist faces is $D(p) = A-p$, where $A > w + r + s$. Suppose that robots are cheap: $r < w + s$.
Essentially, we're asked to find the optimal levels (yields lowest costs) of input in terms of q for H,L,K
I initially thought since $r<w+s$, and $w<s$, we can deduce that $r<s+s$, and for this reason, the production function would give either $q = H+K$, $q = L+H$, or $q = L+K$ (making $H+H$ the only impossible combination). However, I'm super unsure if this is the right thought process, and I don't know how to proceed from here.
Any help is greatly appreciated.