, 2025

In a room, there are $n$ lights numbered with positive integers $1$, $2$, $\ldots$, $n$. At the beginning of the game subsets $S_1$, $S_2$, $\ldots$, $S_k$ of $\{1,2,\ldots,n\}$ can be chosen. For every integer $1\leq i\leq k$, there is a button that turns on the lights corresponding to the elements of $S_i$ and also a button that turns off all the lights corresponding to the elements of $S_i$. For any positive integer $n$, determine the smallest $k$ for which it is possible to choose the sets $S_1$, $S_2$, $\ldots$, $S_n$ in such a way that allows any combination of the $n$ lights to be turned on, starting from the state where all the lights are off.