FKMO 2019 Problem 1

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 $L(3k)=k$ .