AMC 10 2021-fall-a Problem 22

Inside a right circular cone with base radius $5$ and height $12$ are three congruent spheres with radius $r$. Each sphere is tangent to the other two spheres and also tangent to the base and side of the cone. What is $r$? $\textbf{(A)}\ \frac{3}{2} \textbf{(B)}\ \frac{90-40\sqrt{3}}{11} \textbf{(C)}\ 2 \textbf{(D)}\ \frac{144-25\sqrt{3}}{44} \textbf{(E)}\ \frac{5}{2}$