
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 all integers  $k\geqslant 1$  with the following property: given  $k$  different colours, if each integer is coloured in one of these  $k$  colours, then there must exist integers  $a_1<a_2<\cdots<a_{2023}$  of the same colour such that the differences  $a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}$  are all powers of  $2$ .

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