The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has \(n\) aluminum 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 \(k \leq 2n\), Gilberty repeatedly performs the following operation: he 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. For example, if \(n=4\) and \(k=4\), the process starting from the ordering \(AABBBABA\) would be \(AABBBABA \to BBBAAABA \to AAABBBBA \to BBBBAAAA \to \ldots\).

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.