The secant passing through the intersection points of the circle with center $O_{1}$ and radius $4 \mathrm{~cm}$ and the circle with center $O_{2}$ and radius $6 \mathrm{~cm}$ intersects the segment $O_{1} O_{2}$ at point $T$, and we know that the length of this segment is not less than $6 \mathrm{~cm}$. The larger circle intersects the segment $O_{1} O_{2}$ at point $A$, and the smaller circle intersects it at point $B$, and $A T: B T=1: 2$. Calculate the length of the segment $O_{1} O_{2}$.