<|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
 Peter and Paul play the following game. First, Peter chooses some positive integer  $a$  with the sum of its digits equal to  $2012$ . Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is  $2012$ . On each of his moves Paul chooses a positive integer  $x$  and Peter tells him the sum of the digits of  $|x - a|$ . What is the minimal number of moves in which Paul can determine Peter’s number for sure? 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
2012
Peter and Paul play the following game. First, Peter chooses some positive integer  $a$  with the sum of its digits equal to  $2012$ . Paul wants to determine this number, he knows only that the sum of the digits of Peter’s number is  $2012$ . On each of his moves Paul chooses a positive integer  $x$  and Peter tells him the sum of the digits of  $|x - a|$ . What is the minimal number of moves in which Paul can determine Peter’s number for sure?
test