
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:
[list=a][*] $m = n = p = 2,$ [*] arbitrary (but fixed) positive real numbers  $m, n, p.$ [/list]

*Stijn Cambie*


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