
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
\[
\dfrac{4}{x-4} + \dfrac{6}{x-6} + \dfrac{18}{x-18} + \dfrac{20}{x-20} = x^2 - 13x - 6
\]
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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