
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 \( p_{0}(x), p_{1}(x), p_{2}(x), \ldots \) be polynomials such that \( p_{0}(x)=x \) and for all positive integers \( n \), \( \frac{d}{dx} p_{n}(x)=p_{n-1}(x) \). Define the function \( p(x):[0, \infty) \rightarrow \mathbb{R} \) by \( p(x)=p_{n}(x) \) for all \( x \in[n, n+1] \). Given that \( p(x) \) is continuous on \( [0, \infty) \), compute
\[ \sum_{n=0}^{\infty} p_{n}(2009) \]

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