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