Extended circular nim

Circular nim $CN(m, k)$ is a variant of nim, in which there are $m$ piles of tokens arranged in a circle and each player, in their turn, chooses at most $k$ consecutive piles in the circle and removes an arbitrary number of tokens from each pile. The player must remove at least one token in total. For some cases of $m$ and $k$, closed formulas to determine which player has a winning strategy have been found. Almost all cases are still open problems. In this paper, we consider a variant of circular nim, extended circular nim. In extended circular nim $ECN(m_S, k)$, there are $m$ piles of tokes arranged in a circle. $S$ is a set of positive integers less than or equal to half of $m$. In each turn, a player chooses an integer $s \in S$. Then the player selects at most $k$ piles among those located every $s$-th position on the circle, and removes an arbitrary number of tokens from each selected pile. We show some closed formulas to determine which player has a winning strategy for the cases where the number of piles is no more than eight, and for a few generalized cases.