
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
\[\dfrac{2}{x-2} + \dfrac{4}{x-4} + \dfrac{18}{x-18} + \dfrac{20}{x-20} = x^2 - 12x - 6\]
There are positive integers $d, e,$ and $f$ such that $n = d + \sqrt{e + \sqrt{f}}$. Find $d+e+f$.

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