Triangle With Random Constraints?

Geometry Level 5

For a particular triangle, the reciprocal of the angles follows a harmonic progression, and two of the side lengths of this triangle are given as 4 and 5. Find the sum of all distinct possible values of the perimeter of this triangle.

\[31+\sqrt{21}\] \[2+\sqrt{13}+\sqrt{21}\] \[20+\sqrt{13}+\sqrt{21}\] \[29+\sqrt{13}+\sqrt{21}\] \[27\] \[63\] None of the other options No such triangle exists