
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 consisting of points inside and on the boundary of a regular hexagon with side length 1. Determine the minimum $r$ such that there exists a three-coloring scheme of the points in $s$ in which the distance between any two points of the same color is less than $r$.

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