
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
\[\frac{4}{x-4} + \frac{6}{x-6} + \frac{18}{x-18} + \frac{20}{x-20} = x^2 - 12x - 5.\]
There are positive integers \( a, b, \) and \( c \) such that \( n = a + \sqrt{b + \sqrt{c}} \). Find \( a+b+c \).

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