
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 integers \( n \) such that there exists an integer \( a \) satisfying \( a^{\frac{\varphi(n)}{2}} \equiv -1 \pmod{n} \) (where \( \varphi \) is Euler's totient function).

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