
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
Consider a convex polyhedron without parallel edges and without an edge parallel to any face other than the two faces adjacent to it. Call a pair of points of the polyhedron *antipodal* if there exist two parallel planes passing through these points and such that the polyhedron is contained between these planes. Let  $A$  be the number of antipodal pairs of vertices, and let  $B$  be the number of antipodal pairs of midpoint edges. Determine the difference  $A-B$  in terms of the numbers of vertices, edges, and faces.

*Proposed by Kei Irei, Japan*

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