<|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
 Find the minimum positive integer  $k$  such that there exists a function  $f$  from the set  $\Bbb{Z}$  of all integers to  $\{1, 2, \ldots k\}$  with the property that  $f(x) \neq f(y)$  whenever  $|x-y| \in \{5, 7, 12\}$ . 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
4
Find the minimum positive integer  $k$  such that there exists a function  $f$  from the set  $\Bbb{Z}$  of all integers to  $\{1, 2, \ldots k\}$  with the property that  $f(x) \neq f(y)$  whenever  $|x-y| \in \{5, 7, 12\}$ .
test