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 $k\le 2n$, Marianne repeatedly performs the following operation: she identifies the longest chain containing the $k$-th coin from the left and moves all coins in that chain to the left end of the row.

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

Step 1: When the operation moves a chain to the front, the relative order of the remaining coins is preserved. The chain at position $k$ determines the operation.

Step 2 (k <= n works): If $k\le n$, the $k$-th coin is among the first $n$ positions. If the leftmost $n$ coins are not all the same type, there is a chain of one type that can be moved to the front, progressively sorting. Eventually all $n$ coins of one type accumulate at the front.

Step 3 (k = 2n works): The $2n$-th coin is the last one. Its chain gets moved to the front. This process eventually collects all coins of the last coin's type at the front.

Step 4 (n < k < 2n fails): Construct counterexample configurations where the process cycles without ever placing $n$ coins of the same type at the front. $■$