Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$

I found one rather interesting but intractable topology problem.

Prove that a torus triangulation cannot have degrees of vertices $5, 7, 6, 6, 6, 6, \ldots$

Despite various attempts to contract the graph or reduce it to irreducible triangulation, nothing came of it.

What can you recommend?

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