
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 a surface composed of 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex, a square, an octagon, and a hexagon meet. How many line segments connecting the vertices of the polyhedron lie inside the polyhedron, rather than on the faces or edges of the polyhedron?

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