
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
Each rational number is painted with one of two colors, white or red. A coloring is called "sanferminera" if for any two rational numbers \( x \) and \( y \) with \( x \neq y \), the following conditions are satisfied:
a) \( xy = 1 \),
b) \( x + y = 0 \),
c) \( x + y = 1 \),

then \( x \) and \( y \) are painted different colors. How many "sanferminera" colorings are there?

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