
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 \( \mathcal{H} \) be a regular hexagon with side length one. Peter picks a point \( P \) uniformly and at random within \( \mathcal{H} \), then draws the largest circle with center \( P \) that is contained in \( \mathcal{H} \). What is the probability that the radius of this circle is less than \( \frac{1}{2} \)?

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