
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 equalities:
\[3(x + y + z) = x^2 + y^2 + z^2\]
and
\[x + y + z = 3.\]
Determine the maximum value $M$ and the minimum value $m$ of $xy + xz + yz$, and compute $M + 10m$.

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