A mixed selection mechanism gives multiples of $k$ two routes into the sample, breaking equal inclusion probability

Consider a population of $N=nk$ units with $k>1$. A sample of size $n$ is obtained by the following method: (i) A unit is drawn from the set of units labelled $\{k,2k,\dots,N\}$ with probability $1/n$ and is included in the sample. (ii) A SRSWOR of size $n-1$ is then obtained from the remaining $N-1$ units in the population and these units are included in the sample. Then: