<|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 square is called *proper* if its sides are parallel to the coordinate axes. Point  $P$  is randomly selected inside a proper square  $S$  with side length 2012. Denote by  $T$  the largest proper square that lies within  $S$  and has  $P$  on its perimeter, and denote by  $a$  the expected value of the side length of  $T$ . Compute  $\lfloor a \rfloor$ , the greatest integer less than or equal to  $a$ . 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
335
A square is called *proper* if its sides are parallel to the coordinate axes. Point  $P$  is randomly selected inside a proper square  $S$  with side length 2012. Denote by  $T$  the largest proper square that lies within  $S$  and has  $P$  on its perimeter, and denote by  $a$  the expected value of the side length of  $T$ . Compute  $\lfloor a \rfloor$ , the greatest integer less than or equal to  $a$ .
test