
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $AABBBABA \to BBBAAABA \to AAABBBBA \to BBBBAAAA \to ...$ 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.

Present the answer in LaTex format: \boxed{Your answer}
user