
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
Find the maximum value of the expression \(x^{2}+y^{2}+z^{2}\), given that \(x\), \(y\), and \(z\) are integers that satisfy the following system of equations:

\[
\left\{\begin{array}{l}
x y + x + y = 20 \\
y z + z + y = 6 \\
x z + x + z = 2
\end{array}\right.
\]

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