<|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 integers $1,2, \ldots, 20$ are arranged in a circle in a certain order. For each integer $k$, Matthieu counts how many integers less than $k$ are among the 9 integers that follow $k$ in the clockwise direction; he counts this as $A(k)$. He also counts how many integers less than $k$ are among the 9 integers that follow $k$ in the counterclockwise direction; he counts this as $B(k)$. Matthieu notes that $A(k)=B(k)$ for all $k$. What is the number diametrically opposite to 11 on the circle? 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
12
The integers $1,2, \ldots, 20$ are arranged in a circle in a certain order. For each integer $k$, Matthieu counts how many integers less than $k$ are among the 9 integers that follow $k$ in the clockwise direction; he counts this as $A(k)$. He also counts how many integers less than $k$ are among the 9 integers that follow $k$ in the counterclockwise direction; he counts this as $B(k)$. Matthieu notes that $A(k)=B(k)$ for all $k$. What is the number diametrically opposite to 11 on the circle?
test