
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 all primes \( p \) such that there exist integers \( a, b, c \), and \( k \) satisfying the equations

\[
\begin{aligned}
& a^{2}+b^{2}+c^{2}=p \\
& a^{4}+b^{4}+c^{4}=k p .
\end{aligned}
\]

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