
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 coins made out of two types of metal: aluminium (denoted A) and copper (denoted C). Morgane has \( n \) aluminium coins, and \( n \) copper coins, and arranges her \( 2n \) coins in a row in some arbitrary initial order. Given a fixed positive integer \( k \leq 2n \), she repeatedly performs the following operation: identify the largest subsequence containing the \( k \)-th coin from the left which consists of consecutive coins made of the same metal, and move all coins in that subsequence to the left end of the row. For example, if \( n = 4 \) and \( k = 4 \), the process starting from the configuration AACCCACA would be
\[ \text{AACCCACA} \rightarrow \text{CCCAAACA} \rightarrow \text{AAACCCCA} \rightarrow \text{CCCCAAAA} \rightarrow \cdots . \]

Find all pairs \((n, k)\) with \( 1 \leq k \leq 2n \) such that for every initial configuration, at some point of the process there will be at most one aluminium coin adjacent to a copper coin.

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