
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 \( S \) be the set of all points on the boundary and the interior of a regular hexagon with a side length of 1. Find the smallest \( r > 0 \) such that there exists a 3-coloring of \( S \) where the distance between any two points of the same color is less than \( r \).

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