
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 colors, if each integer is colored in one of these \( k \) colors, then there must exist integers \( a_{1} < a_{2} < \cdots < a_{2023} \) of the same color such that the differences \( a_{2} - a_{1}, a_{3} - a_{2}, \ldots, a_{2023} - a_{2022} \) are all powers of 2.

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