
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
Determine the number of pairs of real numbers, \((x, y)\), with \(0 \leq x \leq \frac{\pi}{8}\) and \(0 \leq y \leq \frac{\pi}{8}\) that satisfy the equation \(\cos ^{6}(1000 x) - \sin ^{6}(1000 y) = 1\).

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