
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
Real numbers $x,$ $y,$ and $z$ satisfy the following equality:
\[3(x + y + z) = x^2 + y^2 + z^2.\]
Let $N$ be the maximum value of $xy + xz + yz$, and let $n$ be the minimum value of $xy + xz + yz$. Find $N + 15n$.

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