
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
For all positive numbers $x$, $y$, $z$, simplify the product 
\[
(x^2+y^2+z^2)^{-1}(x^{-2}+y^{-2}+z^{-2})(x^2y^2+y^2z^2+z^2x^2)^{-1}[(x^2y^2)^{-1}+(y^2z^2)^{-1}+(z^2x^2)^{-1}]
\]
**A)** $\frac{1}{x^3y^3z^3}$  
**B)** $\frac{1}{x^2y^2z^2}$  
**C)** $\frac{1}{x^4y^4z^4}$  
**D)** $\frac{1}{x+y+z}$  
**E)** $\frac{1}{x^2+y^2+z^2}$

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