
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
Determine the smallest positive integer  $k{}$  satisfying the following condition: For any configuration of chess queens on a  $100 \times 100$  chequered board, the queens can be coloured one of  $k$  colours so that no two queens of the same colour attack each other.

*Russia, Sergei Avgustinovich and Dmitry Khramtsov*

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