
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 on a table, each marked with a unique positive integer ranging from \( 1 \) to \( n \). Initially, the cards are stacked in increasing order from top to bottom. Operation \( A \) involves moving the top card to the bottom of the stack, and operation \( B \) involves removing the top card from the stack. The sequence of operations \( A B B A B B A B B \cdots \) (where \( A \) is followed by two \( B \)'s repeatedly) is performed until only one card remains. Let \( L(n) \) be the number on the remaining card. Find all positive integers \( k \) such that
\[ L(3k) = k. \]

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