
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
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function that satisfies the functional equation
\[ f(x-y)=2009 f(x) f(y) \]
for all \( x, y \in \mathbb{R} \). If \( f(x) \) is never zero, what is \( f(\sqrt{2009}) \)?

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