
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
Define the sequence \(\left\{x_{i}\right\}_{i \geq 0}\) by \(x_{0}=2009\) and \(x_{n}=-\frac{2009}{n} \sum_{k=0}^{n-1} x_{k}\) for all \(n \geq 1\). Compute the value of \(\sum_{n=0}^{2009} 2^{n} x_{n}\).

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