Screw discrete dynamical systems and their applications to exact slow NIM

Given integers $n,k,\mathrm{\ell }$ such that $0<k<n,1<\mathrm{\ell }$ and an integer vector $x=(x_1,\dots ,x_n)$; denote by $m=m(x)$ the number of entries of $x$ that are multiple of $\mathrm{\ell }$. Choose $n-k$ entries of $x$ as follows: if $n-k\le m(x)$, take $n-k$ smallest entries of $x$ multiple of $\mathrm{\ell }$; if $n-k>m(x)$, take all $m$ such entries, if any, and add remaining $n-k-m$ entries arbitrarily, for example, take the largest ones. In one step, the chosen $n-k$ entries (bears) keep their values, while the remaining $k$ (bulls) are reduced by 1. Repeat such steps getting the sequence $S=S(n,k,\mathrm{\ell },x^0)=(x^0\to x^1\to \dots \to x^j\to \dots )$. It is quasi-periodic". More precisely, there is a function $N = N(n,k,\ell,x^0)$ such that for all $j \geq N$ we have $m(x^j) \geq n-k$ and $range(x^j) \leq \ell$, where $range(x) = (\max(x_i \mid i \in [n]) - \min(x_i \mid i \in [n])$. Furthermore, $N$ is a polynomial in $n,k,\ell,$ and $range(x^0)$ and can be computed in time linear in $n,k,\ell$, and $\log(1 + range(x^0))$. After $N$ steps, the system moves like a screw". Assuming that $x_1\le \cdots \le x_n$, introduce the cyclical order on $[n]=\{1,\dots ,n\}$ considering 1 and $n$ as neighbors. Then, bears and bulls partition $[n]$ into two intervals, rotating by the angle $2\pi k\mathrm{/}n$ with every $\mathrm{\ell }$ steps. Furthermore, after every $p=\mathrm{\ell }n\mathrm{/}GCD(n,k)=\mathrm{\ell }LCM(n,k)\mathrm{/}k$ steps all entries of $x$ are reduced by the same value $\delta =pk\mathrm{/}n$, that is, $x_i^{j+p}-x_i^j=\delta$ for all $i\in [n]$ and $j\ge N$. We provide an algorithm computing $N$ (and $x^j$) in time linear in $n,k,\mathrm{\ell },\log (1+range(x^0))$ (and $\log (1+j)$). In case $k=n-1$ and $\mathrm{\ell }=2$ such screw dynamical system are applicable to impartial games.