
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
A regular dodecahedron has 20 vertices and 30 edges. Each vertex is the intersection point of 3 edges. The other endpoints of these 3 edges are 3 other vertices of the dodecahedron, which we consider to be adjacent to this vertex. Place a real number at each vertex such that the number at each vertex is exactly the arithmetic mean of the numbers at its 3 adjacent vertices. Let $M$ and $m$ be the maximum and minimum of these 20 numbers, respectively. What is the range of values for $M - m$?

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