
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 that $f(x)$ is a function defined on $\mathbb{R}$, and for any real number $x$, it holds that $f(x+4) \leq f(x) + 4$ and $f(x+2) \geq f(x) + 2$, and $f(1) = 0$, determine the value of $f(2009)$.

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