
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 $\mathrm{S}$ be the set of points inside and on the boundary of a regular hexagon with side length 1. Find the minimum $\mathrm{r}$ such that there exists a three-coloring of the points in $\mathrm{S}$, where the distance between any two points of the same color is less than $\mathrm{r}$.

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