
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
Let $x,$ $y,$ $z,$ be nonzero real numbers such that $x + y + z = -1.$  Find all possible values of
\[\frac{x^2 y^2}{(x^2 - yz)(y^2 - xz)} + \frac{x^2 z^2}{(x^2 - yz)(z^2 - xy)} + \frac{y^2 z^2}{(y^2 - xz)(z^2 - xy)}.\]

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