
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
Determine all functions \( f:(0,+\infty) \rightarrow \mathbb{R} \) such that \( f(2009) = 1 \) and
\[ f(x)f(y) + f\left(\frac{2009}{x}\right)f\left(\frac{2009}{y}\right) = 2f(xy) \]
for all positive real numbers \( x \) and \( y \).

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