
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
Given the function
\[ f(x) = \begin{cases} 
\frac{3+x}{1+x^2}, & 0 \leq x \leq 3, \\ 
\frac{3}{5}, & x > 3 
\end{cases} \]
and the sequence \(\{a_n\}\) satisfying \(0 < a_n \leq 3\) for \(n \in \mathbb{N}^*\) and \(\sum_{i=1}^{2009} a_i = \frac{2009}{3}\), if the inequality 
\[
\sum_{i=1}^{2009} f(a_i) \leq x - \ln(x - p)
\]
holds for all \(x \in (p, +\infty)\), find the minimum value of the real number \(p\).

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