
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
We randomly throw a circular metal disk onto a floor made of equilateral triangular tiles. What should be the diameter of the metal disk so that the probability of the disk falling entirely within one triangle is equal to the probability of the disk overlapping two triangle sides but not covering any vertices? (Assume the side length of the triangles is one unit.)

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