Merging $\lim^1 \mathbf{A} \ne 0$ with other nonvanishing constructions

We develop methods for forcing $\lim^1 \mathbf{A} \ne 0$, where $\mathbf{A}$ is a particular inverse system of abelian groups introduced by Marde\v{s}i\'c and Prasolov in their computation of certain strong homology groups. These methods allow us to extend previous nonvanishing results of Casarosa and Lambie-Hanson for $\lim^k \mathbf{A}$ for $k \geq 2$. Specifically we show that, for a given $n$, it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_n$ and $\lim^k \mathbf{A} \ne 0$ whenever $1 \leq k \leq n$ (previously established with $2 \leq k \leq n$). We also show it is relatively consistent with ZFC that $\mathfrak{b} = \mathfrak{d} = \omega_{\omega+2}$ and $\lim^k \mathbf{A} \ne 0$ for all $k \geq 1$ (previously established with $k \geq 2$). We also adapt proofs of Kamo to show that $\lim^1 \mathbf{A} = 0$ holds in many finite support iterated forcing extensions.