
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
In the given equations, there are square roots of \(x, y,\) and \(z\), as well as fractions containing \(x, y,\) and \(z\) in the denominator. Therefore, the variables must be positive. Multiply all three equations by \(\sqrt{2}\) and add them together.

\[
\sqrt{2} x+\frac{1}{\sqrt{2} x}+\sqrt{2} y+\frac{1}{\sqrt{2} y}+\sqrt{2} z+\frac{1}{\sqrt{2} z} = 6-2 \sqrt{2 x}|y-z|-\sqrt{2 y}(x-z)^{2}-\sqrt{2 z} \sqrt{|x-y|}
\]

The right side of the obtained equality does not exceed 6, and equality is achieved only if \(x = y = z\). On the left side, we have the sum \(f(x) + f(y) + f(z)\), where the function \(f\) is defined by the equation \(f(t) = \sqrt{2} t + \frac{1}{\sqrt{2} t}\). This function on the positive half-axis has a minimum, equal to 2, at \(t = \frac{1}{\sqrt{2}}\). Therefore, the value on the left side is not less than 6. Equality can only be achieved under the condition \(x = y = z = \frac{1}{\sqrt{2}}\).

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