
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
Given the real numbers \( x, y, \) and \( z \) that satisfy \( xy \neq 0 \) and \( x+y \neq 0 \), find the minimum value of the expression
$$
\left(\frac{y+z}{x}+2\right)^{2}+\left(\frac{z}{y}+2\right)^{2}+\left(\frac{z}{x+y}-1\right)^{2}.
$$

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