
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  $x,y,z$  are chosen such  that   $$ \frac{1}{|x^2+2yz|} ,\frac{1}{|y^2+2zx|} ,\frac{1}{|x^2+2xy|}  $$   
are lengths of a non-degenerate triangle .
Find all possible values of  $xy+yz+zx$  .

*Proposed by Michael Rolínek*

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