<|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
 A triangle  $ABC$  with  $AC=20$  is inscribed in a circle  $\omega$ . A tangent  $t$  to  $\omega$  is drawn through  $B$ . The distance  $t$  from  $A$  is  $25$  and that from  $C$  is  $16$ .If  $S$  denotes the area of the triangle  $ABC$ , find the largest integer not exceeding  $\frac{S}{20}$  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
10
A triangle  $ABC$  with  $AC=20$  is inscribed in a circle  $\omega$ . A tangent  $t$  to  $\omega$  is drawn through  $B$ . The distance  $t$  from  $A$  is  $25$  and that from  $C$  is  $16$ .If  $S$  denotes the area of the triangle  $ABC$ , find the largest integer not exceeding  $\frac{S}{20}$ 
test