
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 points inside and on the edges of a regular hexagon with side length 1. Find the smallest $r$ such that there exists a 3-coloring of the points in $S$ such that the distance between any two points of the same color is less than $r$.

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