
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 the function $f(x)$, defined on the natural numbers $N$, such that $f(x) = f(x-1) + f(x+1)$ for all $x \in N^{*}$, and given that $f(1) = 2009$ and $f(3) = 0$, determine the number of distinct values that $f(x)$ can take.

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