
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
Determine the largest real solution \( n \) to the equation
\[
\frac{4}{x-4} + \frac{6}{x-6} + \frac{18}{x-18} + \frac{20}{x-20} = x^2 - 13x - 6
\]
Given that \( n \) can be expressed as \( d + \sqrt{e + \sqrt{f}} \) where \( d, e, \) and \( f \) are positive integers, find \( d+e+f \).

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