BALKAN-SL 2020 C2

Let $k$ be a positive integer. Determine the least positive integer $n$, with $n\geq k+1$, for which the game below can be played indefinitely:

Consider $n$ boxes, labelled $b_1,b_2,...,b_n$. For each index $i$, box $b_i$ contains exactly $i$ coins. At each step, the following three substeps are performed in order: (1) Choose $k+1$ boxes; (2) Of these $k+1$ boxes, choose $k$ and remove at least half of the coins from each, and add to the remaining box, if labelled $b_i$, a number of $i$ coins. (3) If one of the boxes is left empty, the game ends; otherwise, go to the next step.

Proposed by Demetres Christofides, Cyprus