TSTST 2025 Problem 5

A tetrahedron $ABCD$ is said to be angelic if it has nonzero volume and satisfies

$$\begin{aligned} \angle BAC + \angle CAD + \angle DAB &= \angle ABC + \angle CBD + \angle DBA, \\ \angle ACB + \angle BCD + \angle DCA &= \angle ADB + \angle BDC + \angle CDA. \end{aligned}$$

Across all angelic tetrahedrons, what is the maximum number of distinct lengths that could appear in the set $\{AB, AC, AD, BC, BD, CD\}$?