
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 equation:
\[ 3(x + y + z) = x^2 + y^2 + z^2. \]
Let \( N \) be the maximum value of \( xy + xz + yz \), and let \( n \) be the minimum value of \( xy + xz + yz \). Find \( N + 5n \).

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