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Color Coding

To help identify destinations, we color the corners like this.

Then when we notice, for example, that the path on the {$3\times 5\times6$} box ends at the yellow corner

we color the point in the box triangle corresponding to this box (distances {$\frac 3{14}$}, {$\frac 5{14}$}, {$\frac 6{14}$}) yellow.