
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
Given a cube with six different positive integers written on its six faces, and the sum of these six integers is 60. An operation can be performed on this cube as follows: each time, select a vertex of the cube and add 1 to the numbers on the three faces meeting at that vertex. After multiple operations, all the numbers on the cube's faces become the same. How many distinct cubes meet this condition? (Once the six numbers are chosen, it is considered one way, regardless of how the numbers are placed on the cube.)

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