
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
Count the number of functions \( f: \mathbb{Z} \rightarrow \{\text{'green','blue'}\} \) such that \( f(x) = f(x+22) \) for all integers \( x \) and there does not exist an integer \( y \) with \( f(y) = f(y+2) = \text{'green'} \).

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