
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 \( x_{i} \in \mathbb{R}^{+} (i = 1, 2, \ldots, 2010) \) and \( \sum_{i=1}^{2010} x_{i}^{2009} = 1 \). Find \( \min \left(\sum_{i=1}^{2010} \frac{x_{i}^{2008}}{1 - x_{i}^{2009}}\right) \) and provide proof.

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