
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
Let \( m, n, p \) be fixed positive real numbers which satisfy \( mnp = 8 \). Depending on these constants, find the minimum of

\[ x^2 + y^2 + z^2 + mxy + nxz + pyz \]

where \( x, y, z \) are arbitrary positive real numbers satisfying \( xyz = 8 \). When is the equality attained? Solve the problem for:
a) \( m = n = p = 2 \),

b) arbitrary (but fixed) positive real \( m, n, p \).

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