<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 The incircle  $\Gamma$  of a scalene triangle  $ABC$  touches  $BC$  at  $D, CA$  at  $E$  and  $AB$  at  $F$ . Let  $r_A$  be the radius of the circle inside  $ABC$  which is tangent to  $\Gamma$  and the sides  $AB$  and  $AC$ . Define  $r_B$  and  $r_C$  similarly. If  $r_A = 16, r_B = 25$  and  $r_C = 36$ , determine the radius of  $\Gamma$ . Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
74
The incircle  $\Gamma$  of a scalene triangle  $ABC$  touches  $BC$  at  $D, CA$  at  $E$  and  $AB$  at  $F$ . Let  $r_A$  be the radius of the circle inside  $ABC$  which is tangent to  $\Gamma$  and the sides  $AB$  and  $AC$ . Define  $r_B$  and  $r_C$  similarly. If  $r_A = 16, r_B = 25$  and  $r_C = 36$ , determine the radius of  $\Gamma$ .
test