
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

Let \( n>2 \) be an integer. Anna, Edda, and Magni play a game on a hexagonal board tiled with regular hexagons, with \( n \) tiles on each side. The game begins with a stone on a tile in one corner of the board. Edda and Magni are on the same team, playing against Anna, and they win if the stone is on the central tile at the end of any player's turn. Anna, Edda, and Magni take turns moving the stone: Anna begins, then Edda, then Magni, and so on.

The rules for each player's turn are:
- Anna has to move the stone to an adjacent tile, in any direction.
- Edda has to move the stone straight by two tiles in any of the 6 possible directions.
- Magni has a choice of passing his turn or moving the stone straight by three tiles in any of the 6 possible directions.

Find all \( n \) for which Edda and Magni have a winning strategy.

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