
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 all pairs of real numbers \( (x ; y) \) that satisfy the inequality \( \sqrt{x+y-1} + x^{4} + y^{4} - \frac{1}{8} \leq 0 \). As an answer, write down the maximum value of the product \( x y \) for all the pairs \( (x ; y) \) found.

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