
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
Inside an equilateral triangle with unit side length, three points are marked such that connecting each of these points to any vertex of the triangle results in lines that divide the angle at that vertex into four equal parts. There are two different sets of distances between the three points calculated by two students, both of which were accepted as correct by the teacher. How is this possible, and what were the results?

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