Question 169
A tetrahedron $ABCD$ is said to be *angelic* if it has nonzero volume and satisfies
$$
\angle BAC + \angle CAD + \angle DAB = \angle ABC + \angle CBD + \angle DBA, \\
\angle ACB + \angle BCD + \angle DCA = \angle ADB + \angle BDC + \angle CDA.
$$
Across all angelic tetrahedrons, what's the maximum number of distinct lengths that could appear in the set $\{AB, AC, AD, BC, BD, CD\}$?
$$
\angle BAC + \angle CAD + \angle DAB = \angle ABC + \angle CBD + \angle DBA, \\
\angle ACB + \angle BCD + \angle DCA = \angle ADB + \angle BDC + \angle CDA.
$$
Across all angelic tetrahedrons, what's the maximum number of distinct lengths that could appear in the set $\{AB, AC, AD, BC, BD, CD\}$?