
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 the largest positive integer \( n \) such that there exist positive integers \( x, y, z \) so that
\[ n^{2} = x^{2} + y^{2} + z^{2} + 2xy + 2yz + 2zx + 3x + 3y + 3z - 6 \]

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