
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For any positive integer  $n$ , let  $a_n$  denote the closest integer to  $\sqrt{n}$ , and let  $b_n=n+a_n$ . Determine the increasing sequence  $(c_n)$  of positive integers which do not occur in the sequence  $(b_n)$ .

Present the answer in LaTex format: \boxed{Your answer}
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