Let \triangle A B C be a triangle with \angle B A C=90^{\circ}, \angle A B C=60^{\circ}, and \angle B C A=30^{\circ} and B C=4. Let the incircle of \triangle A B C meet sides B C, C A, A B at points A_{0}, B_{0}, C_{0}, respectively. Let \omega_{A}, \omega_{B}, \omega_{C} denote the circumcircles of triangles \triangle B_{0} I C_{0}, \triangle C_{0} I A_{0}, \triangle A_{0} I B_{0}, respectively. We construct triangle T_{A} as follows: let A_{0} B_{0} meet \omega_{B} for the second time at A_{1} \neq A_{0}, let A_{0} C_{0} meet \omega_{C} for the second time at A_{2} \neq A_{0}, and let T_{A} denote the triangle \triangle A_{0} A_{1} A_{2}. Construct triangles T_{B}, T_{C} similarly. If the sum of the areas of triangles T_{A}, T_{B}, T_{C} equals \sqrt{m}-n for positive integers m, n, find m+n.