TopsOJ Summer 2025 - Team Round - Problem 12

Triangle ABC has incircle \omega centered at I with radius 5 and A-excircle \omega_A centered at I_A with radius 13. Denote the point where \omega and \omega_A are tangent to \overline{BC} as D and E, respectively. Furthermore, denote the intersection of \overline{II_A} with the circumcircle of \triangle ABC as X. Given that DE=8, find XB \cdot XC.