
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 \( R \) be the region consisting of all points inside or on the boundary of a given circle of radius 1. Find, with proof, all positive real numbers \( d \) such that it is possible to color each point of \( R \) red, green, or blue such that any two points of the same color are separated by a distance less than \( d \).

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