
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
Hudson labels each of the four vertices of a triangular pyramid with a different integer chosen from 1 to 15. For each of the four triangular faces, he then calculates the mean of the three numbers at the vertices of the face. Given that the means calculated by Hudson are all integers, how many different sets of four numbers could he have chosen to label the vertices of the triangular pyramid?

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