AIME 2026a Problem 12

Triangle $\triangle ABC$ lies in plane $\mathcal{P}$ with $AB=6$, $AC=4$, and $\angle BAC=90^{\circ}$. Let $D$ be the reflection across $\overline{BC}$ of the centroid of $\triangle ABC$. Four spheres, all on the same side of $\mathcal{P}$, have radii $1$, $2$, $3$, and $r$ and are tangent to $\mathcal{P}$ at points $A$, $B$, $C$, and $D$, respectively. The four spheres are also each tangent to a second plane $\mathcal{T}$ and are all on the same side of $\mathcal{T}$. The value of $r$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.