Let $m$ and $n$ be positive integers. Kellem and Carmen play the following game: initially, the number $0$ is on the board. Starting with Kellem and alternating turns, they add powers of $m$ to the previous number on the board, such that the new value on the board does not exceed $n$. The player who writes $n$ wins. Determine, for each pair $(m, n)$, who has the winning strategy. Note: A power of $m$ is a number of the form $m^k$, where $k$ is a non-negative integer.