<|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
 Gus has to make a list of  $250$  positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if  $15$  is a number from the list so the list contains  $15$  numbers other than  $15$ . Determine the maximum number of distinct numbers the Gus list can contain. 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
21
Gus has to make a list of  $250$  positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if  $15$  is a number from the list so the list contains  $15$  numbers other than  $15$ . Determine the maximum number of distinct numbers the Gus list can contain.
test