
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 \[
n^2 = x^2 + y^2 + z^2 + 2xy + 2yz + 2zx + 6x + 6y + 6z - 18.
\]

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