Let $n\geq 2$ be a positive integer and $k$ be a number between $1$ and $n$. Recently, I came across the following question about $\mathbb Z/n\mathbb Z$ and I wonder if it was studied before. I'd be thankful for any references or any suggestions on how to approach the problem.
Find the smallest integer $t$ (or an upper bound on $t$) such that, no matter which $t$ distinct non-zero elements $i_1,i_2,\ldots,i_t \in \mathbb Z/n\mathbb Z$ you pick, each element $j\in \mathbb Z/n\mathbb Z$ can be written in the form
$$ j \equiv \ell i_r \pmod n, $$
where $0\leq \ell \leq k$ and $1\leq r \leq t$. In other words, $t$ $(k+1)$-term arithmetic progressions of the form $0, i_r, 2i_r, \ldots, ki_r$ entirely cover $\mathbb Z/n\mathbb Z$.
I am especially interested in the case when $n$ is prime and $k \approx n/2$.