Holiday Contest 2025 - Guts Round - Set 2 Problem 2

In \triangle ABC, \dfrac{\cos{A}}{BC} = \dfrac{\cos{B}}{AC} = \dfrac{\cos{C}}{AB}. Given that side BC = 4, then the area of \triangle ABC can be expressed as a\sqrt{b}, where a and b are relatively prime positive integers and b is not divisible by the square of any prime. Compute a+b.