
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]


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If a prime number \( p \) is a divisor of some number of the form \( a^2 + 3 \), then \( p \) is either equal to 2 or 3, or has the form \( 3k + 1 \).

1) For example, numbers of such form include \( 11 = 3 \cdot 3 + 2 \) and \( 17 = 3 \cdot 5 + 2 \). Conversely, prime numbers 7 and 13 have the form \( 3k + 1 \).

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