
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 convex polyhedron has 12 square faces, 8 regular hexagonal faces, and 6 regular octagonal faces. Exactly one square, one hexagon, and one octagon meet at each vertex of the polyhedron. How many segments joining pairs of vertices of the polyhedron are interior to the polyhedron, that is, are not edges nor contained in a face?

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