
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 largest possible real number  $C$  such that for all pairs  $(x, y)$  of real numbers with  $x \ne y$  and  $xy = 2$ ,   $$ \frac{((x + y)^2- 6))(x-y)^2 + 8))}{(x-y)^2} \ge C. $$   Also determine for which pairs  $(x, y)$  equality holds.

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