
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
In how many ways can the set of ordered pairs of integers be colored red and blue such that for all \(a\) and \(b\), the points \((a, b)\), \((-1-b, a+1)\), and \((1-b, a-1)\) are all the same color?

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