AIME 2026b Problem 12

Consider a tetrahedron with two isosceles triangle faces with side lengths $5\sqrt{10}, 5\sqrt{10},$ and $10$ and two isosceles triangle faces with side lengths $5\sqrt{10}, 5\sqrt{10},$ and $18.$ The four vertices of the tetrahedron lie on a sphere with center $S,$ and the four faces of the tetrahedron are tangent to a sphere with center $R.$ The distance $RS$ can be written as $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find ${}m + n.$ [asy] size(320); pair S = (0, 0); pair A = (0, 110); pair D = (-95, -55); pair C = (95, -55); pair B = (-25, 15); pair R = (0, 25); draw(circle(S, 110), gray+linewidth(0.6)); draw(circle(R, 43), gray+linewidth(0.4)); draw(A--C, black+linewidth(1.1)); draw(A--D, black+linewidth(1.1)); draw(C--D, black+linewidth(1.1)); draw(A--B, dashed+gray+linewidth(0.9)); draw(B--C, dashed+gray+linewidth(0.9)); draw(B--D, dashed+gray+linewidth(0.9)); draw(S--R, dotted+black+linewidth(1.3)); dot(A, p=black+linewidth(4.5)); dot(B, p=black+linewidth(4.5)); dot(C, p=black+linewidth(4.5)); dot(D, p=black+linewidth(4.5)); dot(S, p=red+linewidth(4.5)); dot(R, p=blue+linewidth(4.5)); label("$A$", A, N3.5); label("$B$", B, NW3.0); label("$C$", C, SE3.5); label("$D$", D, SW3.5); label("$S$", S, SW2.0); label("$R$", R, NE2.0); label("$10$", (A+B)/2, NE2.5); label("$10$", (C+D)/2, S3.0); label("$18$", (A+D)/2, NW3.0); label("$18$", (B+C)/2, N2.5); label("$5\sqrt{10}$", (A+C)/2, NE3.0); label("$5\sqrt{10}$", (B+D)/2, S3.0); [/asy]