<|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
 Let triangle \( A B C \) have incircle \(\omega\), which touches \(BC\), \(CA\), and \(AB\) at \(D\), \(E\), and \(F\), respectively. Then, let \(\omega_{1}\) and \(\omega_{2}\) be circles tangent to \(AD\) and internally tangent to \(\omega\) at \(E\) and \(F\), respectively. Let \(P\) be the intersection of line \(EF\) and the line passing through the centers of \(\omega_{1}\) and \(\omega_{2}\). If \(\omega_{1}\) and \(\omega_{2}\) have radii 5 and 6, respectively, compute \(PE \cdot PF\). 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
3600
Let triangle \( A B C \) have incircle \(\omega\), which touches \(BC\), \(CA\), and \(AB\) at \(D\), \(E\), and \(F\), respectively. Then, let \(\omega_{1}\) and \(\omega_{2}\) be circles tangent to \(AD\) and internally tangent to \(\omega\) at \(E\) and \(F\), respectively. Let \(P\) be the intersection of line \(EF\) and the line passing through the centers of \(\omega_{1}\) and \(\omega_{2}\). If \(\omega_{1}\) and \(\omega_{2}\) have radii 5 and 6, respectively, compute \(PE \cdot PF\).
test