Christmas Contest - Team Round - Problem 16

Let \omega be a circle with radius 5 and let \Gamma be a circle with diameter 1 that is internally tangent to \omega at point T. Let AB be a chord of \omega that is tangent to \Gamma at point C such that AB=8. Given that TC^2=\frac{m}{n} for relatively prime positive integers m and n, find m+n.