
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
Determine all natural numbers  $n$  such that for each natural number  $a$  relatively prime with  $n$  and  $a \le 1 + \left\lfloor \sqrt{n} \right\rfloor$  there exists some integer  $x$  with  $a \equiv x^2 \mod n$ .

Remark: "Natural numbers" is the set of positive integers.

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