
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
In ZS Chess, an Ivanight attacks like a knight, except that if the attacked square is out of range, it goes
through the edge and comes out from the other side of the board, and attacks that square instead. The
ZS chessboard is an  $8 \times 8$  board, where cells are coloured with  $n$  distinct colours, where  $n$  is a natural
number, such that a Ivanight placed on any square attacks  $ 8 $  squares that consist of all  $n$  colours, and
the colours appear equally many times in those  $ 8 $  squares. For which values of  $n$  does such a ZS chess
board exist?

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