<|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
 Consider the sequence of numbers  $\left[n+\sqrt{2n}+\frac12\right]$ , where  $[x]$  denotes the greatest integer not exceeding  $x$ . If the missing integers in the sequence are  $n_1<n_2<n_3<\ldots$  find  $n_{12}$  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
21
Consider the sequence of numbers  $\left[n+\sqrt{2n}+\frac12\right]$ , where  $[x]$  denotes the greatest integer not exceeding  $x$ . If the missing integers in the sequence are  $n_1<n_2<n_3<\ldots$  find  $n_{12}$ 
test