
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
Find the greatest real number  $C$  such that, for all real numbers  $x$  and  $y \neq x$  with  $xy = 2$  it holds that
\[\frac{((x + y)^2 - 6)((x - y)^2 + 8)}{(x-y)^2}\geq C.\]
When does equality occur?

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