Find the smallest positive integer $n$ such that there exists $r \in (0, \tfrac{1}{1000})$ such that the number $(n+r)^3$ is an integer.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the smallest positive integer $n$ such that there exists $r \in (0, \tfrac{1}{1000})$ such that the number $(n+r)^3$ is an integer.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
19
Find the smallest positive integer $n$ such that there exists $r \in (0, \tfrac{1}{1000})$ such that the number $(n+r)^3$ is an integer.
train