International Mathematical Olympiad 2022 Problem 1

The Bank of Oslo issues two types of coin: aluminium (denoted $A$) and bronze (denoted $B$). Marianne has $n$ aluminium coins and $n$ bronze coins, arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \leq 2n$, Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k^{\text{th}}$ coin from the left, and moves all coins in that chain to the left end of the row. For example, if $n = 4$ and $k = 4$, the process starting from the ordering $AABBBABA$ would be

$$AAB\underline{B}BABA \rightarrow BBB\underline{A}AABA \rightarrow AAA\underline{B}BBBA \rightarrow BBB\underline{B}AAAA \rightarrow BBB\underline{B}AAAA \rightarrow \cdots.$$

Find all pairs $(n,k)$ with $1 \leq k \leq 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.