
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 $n$ be the largest real solution to the equation
\[\dfrac{4}{x-2} + \dfrac{6}{x-6} + \dfrac{13}{x-13} + \dfrac{15}{x-15} = x^2 - 7x - 6\]
There are positive integers $p, q,$ and $r$ such that $n = p + \sqrt{q + \sqrt{r}}$. Find $p+q+r$.

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