
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A point \( D \) is chosen on the side \( BC \) of the triangle \( ABC \). The line tangent to the incircles of the triangles \( ABD \) and \( ADC \) and different from \( BC \) and \( AD \) intersects the segment \( AD \) at \( T \). If \( AB = 40 \ \mathrm{cm}, AC = 50 \ \mathrm{cm} \) and \( BC = 60 \ \mathrm{cm} \), determine the length of \( AT \).

Present the answer in LaTex format: \boxed{Your answer}
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