
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 $f(x) = \frac{x+3}{x^2+1}$, and $g(x) = x - \ln(x-p)$.
(I) Find the equation of the tangent line to the graph of $f(x)$ at the point $\left(\frac{1}{3}, f\left(\frac{1}{3}\right)\right)$;
(II) Determine the number of zeros of the function $g(x)$, and explain the reason;
(III) It is known that the sequence $\{a_n\}$ satisfies: $0 < a_n \leq 3$, $n \in \mathbb{N}^*$, and $3(a_1 + a_2 + \ldots + a_{2015}) = 2015$. If the inequality $f(a_1) + f(a_2) + \ldots + f(a_{2015}) \leq g(x)$ holds for $x \in (p, +\infty)$, find the minimum value of the real number $p$.

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