
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  $M$  be a set of six distinct positive integers whose sum is  $60$ . These numbers are written on the faces of a cube, one number to each face. A *move* consists of choosing three faces of the cube that share a common vertex and adding  $1$  to the numbers on those faces. Determine the number of sets  $M$  for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.

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