
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
There are  $n$  cards such that for each  $i=1,2, \cdots n$ , there are exactly one card labeled  $i$ . Initially the cards are piled with increasing order from top to bottom. There are two operations:



- $A$  : One can take the top card of the pile and move it to the bottom;
- $B$  : One can remove the top card from the pile.

The operation  $ABBABBABBABB \cdots $  is repeated until only one card gets left. Let  $L(n)$  be the labeled number on the final pile. Find all integers  $k$  such that  $L(3k)=k$ .

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