
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
Find the positive integers  $n$  with  $n \geq 4$  such that  $[\sqrt{n}]+1$  divides  $n-1$  and  $[\sqrt{n}]-1$  divides  $n+1$ .
<details><summary>Remark</summary>This problem can be solved in a similar way with the one given at [Cono Sur Olympiad 2006](http://www.mathlinks.ro/Forum/resources.php?c=1&cid=97&year=2006), problem 5.</details>

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