
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
Non-negative numbers \( x, y, z \) satisfy the inequalities \( 5 \leq x, y, z \leq 8 \).

What is the maximum and minimum value that the quantity \( S = 2x^2y^2 + 2x^2z^2 + 2y^2z^2 - x^4 - y^4 - z^4 \) can take?

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