Mock Euclid 2025 - Problem 8 Part A

Acute triangle ABC has inradius 5 and AB < AC. The foot of the altitude from A to BC intersects BC at P, and AP intersects the incircle at points X and Y, with AX<AY. Given that XY = 8 and BP=4, the area of ABC can be written as \frac{m}{n} for relatively prime positive integers m and n. Compute m+n.