
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](https://artofproblemsolving.com/wiki/index.php/Convex) [polyhedron](https://artofproblemsolving.com/wiki/index.php/Polyhedron) has for its [faces](https://artofproblemsolving.com/wiki/index.php/Face) 12 [squares](https://artofproblemsolving.com/wiki/index.php/Square), 8 [regular](https://artofproblemsolving.com/wiki/index.php/Regular_polygon) [hexagons](https://artofproblemsolving.com/wiki/index.php/Hexagon), and 6 regular [octagons](https://artofproblemsolving.com/wiki/index.php/Octagon). At each [vertex](https://artofproblemsolving.com/wiki/index.php/Vertex) of the polyhedron one square, one hexagon, and one octagon meet. How many [segments](https://artofproblemsolving.com/wiki/index.php/Segment) joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an [edge](https://artofproblemsolving.com/wiki/index.php/Edge) or a [face](https://artofproblemsolving.com/wiki/index.php/Face)?

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