
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
Find all continuous functions \( f: \mathbb{R}_{+}^{*} \rightarrow [1, +\infty) \) such that for all \( (n, x) \in \mathbb{N} \times \mathbb{R}_{+}^{*} \):

\[ 
\prod_{k=1}^{n} f(kx) < 2010 n^{2010} 
\]

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