A point $D$ is chosen on side $B C$ of triangle $A B C$. The line tangent to the incircles of triangles $A B D$ and $A D C$ and different from $B C$ and $A D$ intersects segment $A D$ at $T$. If $A B=40 \text{ cm}, A C=50 \text{ cm}$, and $B C=60 \text{ cm}$, determine the value of the length of $A T$.

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