
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 that $$(x+y+z)(xy+xz+yz)=36$$ and that $$x^2(y+z)+y^2(x+z)+z^2(x+y)=12$$ for real numbers $x$, $y$, and $z$, and also given that $$(x+y+z)^2=x^2+y^2+z^2+12,$$ what is the value of $xyz$?

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