TxO Math Bowl 2024 - Individuals A - Problem 13

Circles \omega_1 and \omega_2 intersect at points X_1 and X_2. A line tangent to circles \omega_1 and \omega_2 at A and B, respectively, is closer to X_1 than X_2. The line through X_1 parallel to AB intersect \omega_1 and \omega_2 at D and C, respectively, different from X_1. E is the intersection of DA and CB. If EC = 12, CD = 18, DE = 10. If the length of X_1X_2 can be expressed as \frac{a\sqrt{b}}{c}, where a, b, and c are positive integers and \gcd(a, c) = 1, find a + b + c.