
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  $x,y,z$  satisfy   $$ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+x+y+z=0 $$   and none of them lies in the open interval  $(-1,1)$ . Find the maximum value of  $x+y+z$ .

*Proposed by Jaromír Šimša*

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