
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

Given an even function \( f: \mathbf{Z} \rightarrow \mathbf{Z} \) that satisfies \( f(1)=1 \), \( f(2008) \neq 1 \), and for any integers \(a\) and \(b\), \( f(a+b) \leq \max \{ f(a), f(b) \} \), what are the possible values of \( f(2009) \)?

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