
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 set of identical regular pentagons, with natural numbers from 1 to 5 written at each vertex as shown in the diagram. The pentagons can be rotated and flipped. They are stacked in a pile (vertex to vertex), and it turns out that the sums of the numbers at each of the five vertices are the same. How many pentagons could be in this stack?

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