
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
The numbers \(x, y\) and \(z\) are given by \(x=\sqrt{12-3 \sqrt{7}}-\sqrt{12+3 \sqrt{7}}, y=\sqrt{7-4 \sqrt{3}} - \sqrt{7+4 \sqrt{3}}\) and \(z=\sqrt{2+\sqrt{3}}-\sqrt{2-\sqrt{3}}\). Find the value of \(x y z\).

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