
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 quadruples \((a, b, c, k)\) of integers such that \(a, b, c\) are prime numbers and \(k\) is a positive integer satisfying

\[ a^2 + b^2 + 16c^2 = 9k^2 + 1 \]

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